"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { h3: "h3", li: "li", ol: "ol", p: "p", span: "span", strong: "strong", table: "table", tbody: "tbody", td: "td", th: "th", thead: "thead", tr: "tr", ul: "ul", ..._provideComponents(), ...props.components }, {ComparisonTable, Definition, English, Explanation, Hinglish, ImportantQA, MCQ, SubTopicHeader, Summary} = _components; if (!ComparisonTable) _missingMdxReference("ComparisonTable", true); if (!Definition) _missingMdxReference("Definition", true); if (!English) _missingMdxReference("English", true); if (!Explanation) _missingMdxReference("Explanation", true); if (!Hinglish) _missingMdxReference("Hinglish", true); if (!ImportantQA) _missingMdxReference("ImportantQA", true); if (!MCQ) _missingMdxReference("MCQ", true); if (!SubTopicHeader) _missingMdxReference("SubTopicHeader", true); if (!Summary) _missingMdxReference("Summary", true); return _jsxs(_Fragment, { children: [_jsx(SubTopicHeader, { id: "5.3", title: "Normal Distribution" }), "\n", _jsxs(Definition, { term: "Normal Distribution", children: [_jsx(English, { children: _jsx(_components.p, { children: "The Normal Distribution (also known as the Gaussian Distribution) is a continuous probability distribution that is completely symmetrical about its mean. It is represented by a bell-shaped curve (the Normal Curve) where data near the mean are more frequent in occurrence than data far from the mean." }) }), _jsx(Hinglish, { children: _jsx(_components.p, { children: "Normal Distribution (jise Gaussian Distribution bhi kehte hain) ek continuous probability distribution hai jo apne mean ke both sides pure tarike se symmetrical hoti hai. Ise ek bell-shaped (ghanti ke aakar ke) curve se represent kiya jata hai, jismein mean ke paas wali values ki frequency sabse zyada hoti hai aur door wali values ki frequency kam hoti jati hai." }) })] }), "\n", _jsxs(Explanation, { children: [_jsxs(English, { children: [_jsx(_components.h3, { children: "Key Properties of the Normal Curve" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Bell-Shaped and Symmetrical:" }), " The curve is bell-shaped and symmetric about the vertical line ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] })] }) }), " (mean)."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Equality of Central Tendencies:" }), " The Mean, Median, and Mode of the distribution are all equal and lie exactly at the center of the curve."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Asymptotic to the X-axis:" }), " The tails of the curve extend infinitely on both directions, asymptotically approaching the x-axis but never touching it."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Total Area is 1:" }), " The total area under the normal curve is exactly ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] }) }) }), " (or ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "100%" })] }) }) }), "). Due to symmetry, the area to the left of the mean is ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0.5" })] }) }) }), " and the area to the right is ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0.5" })] }) }) }), "."] }), "\n"] }), _jsxs(_components.h3, { children: ["Standard Normal Variable (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" })] }) }) }), ")"] }), _jsxs(_components.p, { children: ["To calculate probabilities for any normal distribution ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.109em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) }) }), ", we transform the continuous variable ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) }) }), " into the ", _jsxs(_components.strong, { children: ["Standard Normal Variable (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" })] }) }) }), ")"] }), " using the relation:\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1994em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8544em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "σ" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "x" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "μ" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] }) }), "\nwhere:"] }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) }) }), " = value of the random variable."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) }) }), " = mean of the distribution."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" })] }) }) }), " = standard deviation of the distribution."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" })] }) }) }), " represents the number of standard deviations ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) }) }), " is away from the mean. The variable ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" })] }) }) }), " follows the Standard Normal Distribution ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.109em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) }) }), " with mean ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] }) }) }), " and standard deviation ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] }) }) }), "."] }), "\n"] }), _jsx(_components.h3, { children: "The Empirical Rule (68-95-99.7 Rule)" })] }), _jsxs(Hinglish, { children: [_jsx(_components.h3, { children: "Normal Curve Ki Main Properties" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Bell-Shaped aur Symmetrical:" }), " Normal curve ghanti ke aakar ka hota hai aur ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] })] }) }), " (mean) ke left aur right side ek jaisa (symmetric) hota hai."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Central Tendencies Barabar:" }), " Is distribution mein Mean, Median, aur Mode teeno ki value same hoti hai aur woh center mein live karti hain."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Asymptotic to X-axis:" }), " Curve ke dono side ke tails infinite tak stretch hote hain, jo x-axis ke close aate hain par use kabhi touch nahi karte."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Total Area 1 Hota Hai:" }), " Curve ke niche ka total area hamesha ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] }) }) }), " (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "100%" })] }) }) }), ") hota hai. Symmetry ki wajah se, center line (", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] }) }), ") ke left side ka area ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0.5" })] }) }) }), " aur right side ka area ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0.5" })] }) }) }), " hota hai."] }), "\n"] }), _jsxs(_components.h3, { children: ["Standard Normal Variable (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" })] }) }) }), ")"] }), _jsxs(_components.p, { children: ["Kisi bhi normal distribution ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0641em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.109em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) }) }), " ko standard form mein convert karne ke liye hum variable ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) }) }), " ko ", _jsxs(_components.strong, { children: ["Standard Normal Variable (", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" })] }) }) }), ")"] }), " mein badalte hain:\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1994em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8544em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0359em" }, children: "σ" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.4461em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "x" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "μ" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] }) }), "\njahan:"] }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] }) }) }), " = random variable ki value."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) }) }), " = distribution ka mean."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" })] }) }) }), " = distribution ka standard deviation."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" })] }) }) }), " variable Standard Normal Distribution ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.109em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) }) }), " ko follow karta hai jiska mean ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] }) }) }), " aur standard deviation ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] }) }) }), " hota hai."] }), "\n"] }), _jsx(_components.h3, { children: "Empirical Rule (Symmetric Limits Areas)" })] }), _jsx(ComparisonTable, { children: _jsxs(_components.table, { children: [_jsx(_components.thead, { children: _jsxs(_components.tr, { children: [_jsx(_components.th, { style: { textAlign: "left" }, children: "Interval (Standard Deviations)" }), _jsx(_components.th, { style: { textAlign: "left" }, children: "Area Under Curve" }), _jsx(_components.th, { style: { textAlign: "left" }, children: "Percentage of Data Included" })] }) }), _jsxs(_components.tbody, { children: [_jsxs(_components.tr, { children: [_jsxs(_components.td, { style: { textAlign: "left" }, children: [_jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] }) }), " (", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] }) }), ")"] }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0.6827" })] }) }) }) }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "68.27%" })] }) }) }) })] }), _jsxs(_components.tr, { children: [_jsxs(_components.td, { style: { textAlign: "left" }, children: [_jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] }) }), " (", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "2" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] }) }), ")"] }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0.9545" })] }) }) }) }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "95.45%" })] }) }) }) })] }), _jsxs(_components.tr, { children: [_jsxs(_components.td, { style: { textAlign: "left" }, children: [_jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8389em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0359em" }, children: "σ" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] }) }), " (", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord", children: "3" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] }) }), ")"] }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0.9973" })] }) }) }) }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "99.73%" })] }) }) }) })] })] })] }) })] }), "\n", _jsxs(Summary, { children: [_jsx(English, { children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Normal distribution deals with continuous variables (height, weight, bulb life, etc.)." }), "\n", _jsx(_components.li, { children: "Z-score transformation normalizes any distribution to a mean of 0 and SD of 1." }), "\n", _jsxs(_components.li, { children: ["Symmetric property implies ", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: 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"katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1389em" }, children: "P" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.044em" }, children: "z" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord mathnormal", 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The curve is bell-shaped and perfectly symmetrical about the mean ($x = \\mu$).\n2. The Mean, Median, and Mode are all equal and lie at the center of the distribution.\n3. The total area under the curve is exactly equal to 1, with 0.5 area on either side of the center.\n4. The curve is asymptotic to the x-axis, extending to infinity in both directions without ever touching the axis.Normal Distribution curve ki char properties:\n1. Curve bell-shaped hota hai aur mean ($x = \\mu$) ke left aur right symmetric hota hai.\n2. Mean, Median, aur Mode ki values barabar hoti hain aur center mein hoti hain.\n3. Curve ke niche ka total area 1 hota hai, aur center line ke dono sides 0.5 area hota hai.\n4. Curve x-axis par asymptote hota hai, dono sides infinite tak jata hai bina axis ko touch kiye." }] } }, { "q": "If the height of 1000 students is normally distributed with mean 165 cm and standard deviation 5 cm, find the standard normal variable z for a student of height 170 cm.", "marks": 2, "a": { "components": [{ "type": "Explanation", "content": "Given:\n- Mean $\\mu = 165$ cm\n- Standard Deviation $\\sigma = 5$ cm\n- Student height $x = 170$ cm\nThe standard normal variable $z$ is given by:\n$$z = \\frac{x - \\mu}{\\sigma}$$\nSubstitute the values:\n$$z = \\frac{170 - 165}{5} = \\frac{5}{5} = 1$$.\nHence, the standard normal variable $z$ is $1$.Diya hai:\n- Mean $\\mu = 165$ cm\n- Standard Deviation $\\sigma = 5$ cm\n- Student height $x = 170$ cm\nStandard normal variable $z$ ka formula:\n$$z = \\frac{x - \\mu}{\\sigma}$$\nValues put karne par:\n$$z = \\frac{170 - 165}{5} = \\frac{5}{5} = 1$$.\nIsliye, $z$ ki value $1$ hai." }] } }, { "q": "In a test on 2000 electric bulbs, the life of a bulb is normally distributed with mean 2040 hours and standard deviation 60 hours. Find the number of bulbs having a life between 2040 hours and 2160 hours. (Given: Area under normal curve from z=0 to z=2 is 0.4772)", "marks": 4, "a": { "components": [{ "type": "Explanation", "content": "Given:\n- Total bulbs $N = 2000$\n- Mean $\\mu = 2040$ hours\n- Standard Deviation $\\sigma = 60$ hours\nWe want to find the probability of bulbs having a life between $x_1 = 2040$ and $x_2 = 2160$:\n$$P(2040 < x < 2160)$$\nConvert $x$ values to $z$ scores:\n- For $x_1 = 2040$: $z_1 = \\frac{2040 - 2040}{60} = 0$.\n- For $x_2 = 2160$: $z_2 = \\frac{2160 - 2040}{60} = \\frac{120}{60} = 2$.\nSo:\n$$P(2040 < x < 2160) = P(0 < z < 2)$$\nGiven that the area from $z = 0$ to $z = 2$ is $0.4772$:\n$$P(0 < z < 2) = 0.4772$$\nTo find the number of bulbs, multiply this probability by the total number of bulbs $N$:\n$$\\text{Number of bulbs} = N \\cdot P(0 < z < 2) = 2000 \\cdot 0.4772 = 954.4 \\approx 954 \\text{ bulbs}$$.Diya hai:\n- Total bulbs $N = 2000$\n- Mean $\\mu = 2040$ hours\n- Standard Deviation $\\sigma = 60$ hours\nHume $x_1 = 2040$ aur $x_2 = 2160$ ke beech ke bulbs ki probability nikalni hai:\n$$P(2040 < x < 2160)$$\n$x$ ko $z$ values mein badlenge:\n- For $x_1 = 2040$: $z_1 = \\frac{2040 - 2040}{60} = 0$.\n- For $x_2 = 2160$: $z_2 = \\frac{2160 - 2040}{60} = 2$.\nIsliye:\n$$P(2040 < x < 2160) = P(0 < z < 2)$$\n$z = 0$ se $z = 2$ tak ka area $0.4772$ diya hai:\n$$P(0 < z < 2) = 0.4772$$\nBulbs ki sankhya nikalne ke liye probability ko total $N=2000$ se multiply karein:\n$$\\text{Number of bulbs} = 2000 \\cdot 0.4772 = 954.4 \\approx 954 \\text{ bulbs}$$." }] } }] }), "\n", _jsx(MCQ, { questions: [{ "id": "2a693309-237d-4bbe-b75f-78d74d4bb996", "q": "What is the shape of the Normal Distribution curve?", "options": ["Skewed right", "U-shaped", "Bell-shaped", "Straight line"], "answer": 2, "hint": "It rises symmetrically to a peak at the center and tails off on both sides.", "explanation": "The probability density function of a normal distribution forms a symmetric, bell-shaped curve." }, { "id": "01195c86-b443-4894-89c1-62b672973123", "q": "For a standard normal distribution, what are the values of mean (mu) and standard deviation (sigma)?", "options": ["mu = 1, sigma = 0", "mu = 0, sigma = 1", "mu = 0, sigma = 0", "mu = 1, sigma = 1"], "answer": 1, "hint": "Standard normal distribution is centered at zero and scaled by unit standard deviation.", "explanation": "By definition, the standard normal variable z has a mean of 0 and a standard deviation of 1." }, { "id": "b13c4365-103b-4412-86b9-6d5ba00da62b", "q": "What is the total area under the Normal Distribution curve?", "options": ["0.5", "10", "1", "Infinite"], "answer": 2, "hint": "The sum of all probabilities in any probability distribution is always equal to this value.", "explanation": "The total area under the probability density curve of any continuous distribution (including normal) is exactly equal to 1." }, { "id": "02ac2d52-9971-4f0f-8bff-1ea6b6350cc7", "q": "What is the formula to convert a normal random variable x into a standard normal variable z?", "options": ["z = (x - mu) / sigma", "z = (mu - x) / sigma", "z = x - mu * sigma", "z = (x - sigma) / mu"], "answer": 0, "hint": "Subtract the mean and divide by the standard deviation.", "explanation": "The standard normal variable z (z-score) is calculated as z = (x - mu) / sigma." }, { "id": "e0f83218-e357-4d71-b5af-74cbac007be6", "q": "If a normal distribution has mean = 50, which parameter represents the center of the curve?", "options": ["Mean", "Median", "Mode", "All of the above"], "answer": 3, "hint": "In a symmetric normal distribution, all three measures of central tendency coincide.", "explanation": "Due to perfect symmetry, the mean, median, and mode are all equal and located at the center of the normal curve." }, { "id": "eb3fa233-5004-406c-b50c-cf5539e57da9", "q": "What percentage of data lies within one standard deviation (+/- 1 sigma) of the mean in a normal distribution?", "options": ["50%", "68.27%", "95.45%", "99.73%"], "answer": 1, "hint": "This is the first interval in the empirical rule.", "explanation": "According to the empirical rule, approximately 68.27% of the total observations lie within one standard deviation of the mean." }, { "id": "dc51069f-2804-4309-809e-b432e567e2c0", "q": "What is the area to the left of the standard normal variable z = 0?", "options": ["0", "1.0", "0.5", "-0.5"], "answer": 2, "hint": "Since the curve is symmetric about z = 0, it divides the total area of 1 into two equal halves.", "explanation": "The curve is symmetric about z = 0, so the area to the left of the center line is exactly half of the total area, which is 0.5." }, { "id": "14c0c858-6a10-4d01-be04-941e5500449d", "q": "If x is normally distributed with mean 100 and SD 10, find the z-score for x = 85.", "options": ["1.5", "-1.5", "-1.0", "1.0"], "answer": 1, "hint": "Substitute x=85, mu=100, and sigma=10 in the z-score formula.", "explanation": "z = (85 - 100) / 10 = -15/10 = -1.5." }, { "id": "a076c098-7071-4a85-bffd-7e9abcb322b8", "q": "In a normal distribution, the curve is asymptotic to the x-axis. What does 'asymptotic' mean?", "options": ["The curve touches the x-axis at z = 3", "The curve cuts the x-axis at multiple points", "The curve approaches the x-axis closely but never touches it", "The curve is perpendicular to the x-axis"], "answer": 2, "hint": "Think of limits as they tend to infinity.", "explanation": "Asymptotic means the curve approaches the x-axis infinitely closely as x goes to infinity or negative infinity, but mathematically it never intersects or touches it." }, { "id": "3653de0d-f621-4aca-9f6b-34bb81480e5e", "q": "If P(0 < z < 1.5) = 0.4332, what is the probability P(z > 1.5)?", "options": ["0.5668", "0.0668", "0.9332", "0.4332"], "answer": 1, "hint": "The total area to the right of z = 0 is 0.5. Note that P(z > 1.5) = P(z > 0) - P(0 < z < 1.5).", "explanation": "The area to the right of z = 0 is 0.5. Since P(z > 1.5) + P(0 < z < 1.5) = 0.5, we have P(z > 1.5) = 0.5 - 0.4332 = 0.0668." }], topicId: "5.3" })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent }; function _missingMdxReference(id, component) { throw new Error("Expected " + (component ? "component" : "object") + " `" + id + "` to be defined: you likely forgot to import, pass, or provide it."); }