{"id":1347,"date":"2026-01-27T15:08:58","date_gmt":"2026-01-27T05:08:58","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1347"},"modified":"2026-01-28T19:47:19","modified_gmt":"2026-01-28T09:47:19","slug":"year12-math-3-4-2-binomial-and-normal-distributions","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1347","title":{"rendered":"Year12 MATH 3-4-2 binomial and normal distributions"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In QLD Year 12 Mathematical Methods Unit 4, binomial and normal distributions are&nbsp;<mark>core components of probability and statistics, focusing on modeling discrete and continuous data to make inferences<\/mark>. Unit 4, Topic 3 typically covers the Binomial Distribution, while Topic 4 covers Continuous Random Variables and the Normal Distribution.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. Binomial Distribution (Discrete)&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The binomial distribution models the number of successes in a fixed number of independent &#8220;Bernoulli trials,&#8221; where each trial has only two outcomes: success or failure.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Key Requirements:<\/strong> Fixed number of trials (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math>), constant probability of success (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>p<\/mi><annotation encoding=\"text\/plain\">p<\/annotation><\/semantics><\/math>) in each trial, and trials must be independent.<\/li>\n\n\n\n<li><strong>Key Formula:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo>(<\/mo><mi>X<\/mi><mo>=<\/mo><mi>x<\/mi><mo>)<\/mo><mo>=<\/mo><mrow><mo>(<\/mo><mfrac linethickness=\"0\"><mi>n<\/mi><mi>x<\/mi><\/mfrac><mo>)<\/mo><\/mrow><msup><mi>p<\/mi><mi>x<\/mi><\/msup><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><msup><mo>)<\/mo><mrow><mi>n<\/mi><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"text\/plain\">cap P open paren cap X equals x close paren equals the 2 by 1 column matrix; n, x end-matrix; p to the x-th power open paren 1 minus p close paren raised to the n minus x power<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Mean &amp; Variance:<\/strong> Mean (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"text\/plain\">mu<\/annotation><\/semantics><\/math>) = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mi>p<\/mi><\/mrow><annotation encoding=\"text\/plain\">n p<\/annotation><\/semantics><\/math>; Variance (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><annotation encoding=\"text\/plain\">sigma squared<\/annotation><\/semantics><\/math>) = <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mi>p<\/mi><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">n p open paren 1 minus p close paren<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Application:<\/strong> Used for problems like &#8220;finding the probability of exactly <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>k<\/mi><annotation encoding=\"text\/plain\">k<\/annotation><\/semantics><\/math> successes in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math> trials&#8221;.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">2. Normal Distribution (Continuous)&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The normal distribution (or Gaussian distribution) is a continuous, symmetric, &#8220;bell-shaped&#8221; curve used to model continuous data, such as heights, IQ scores, or measurement errors.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Characteristics:<\/strong> Symmetric around the mean (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"text\/plain\">mu<\/annotation><\/semantics><\/math>), where the mean, median, and mode are equal. It is defined by its mean (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"text\/plain\">mu<\/annotation><\/semantics><\/math>) and standard deviation (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>\u03c3<\/mi><annotation encoding=\"text\/plain\">sigma<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Empirical Rule (68-95-99.7% Rule):<\/strong> Approximately 68% of data lies within <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mi>\u03c3<\/mi><\/mrow><annotation encoding=\"text\/plain\">1 sigma<\/annotation><\/semantics><\/math> of the mean, 95% within <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>\u03c3<\/mi><\/mrow><annotation encoding=\"text\/plain\">2 sigma<\/annotation><\/semantics><\/math>, and 99.7% within <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mi>\u03c3<\/mi><\/mrow><annotation encoding=\"text\/plain\">3 sigma<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Standard Normal Distribution (\ud835\udc4d):<\/strong> A special case where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03bc<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">mu equals 0<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c3<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">sigma equals 1<\/annotation><\/semantics><\/math>. The formula <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Z<\/mi><mo>=<\/mo><mfrac><mrow><mi>X<\/mi><mo>\u2212<\/mo><mi>\u03bc<\/mi><\/mrow><mi>\u03c3<\/mi><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">cap Z equals the fraction with numerator cap X minus mu and denominator sigma end-fraction<\/annotation><\/semantics><\/math> is used to calculate probabilities.<\/li>\n\n\n\n<li><strong>Technology Use:<\/strong> CAS calculators are used to calculate probabilities (e.g., <code>normCdf<\/code>) and quantiles (inverse normal).&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">3. Normal Approximation to the Binomial&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In Unit 4, you also learn that for large sample sizes (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math>), a binomial distribution can be approximated by a normal distribution.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Condition:<\/strong> Generally used when <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mi>p<\/mi><mo>&gt;<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"text\/plain\">n p is greater than 5<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo>)<\/mo><mo>&gt;<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"text\/plain\">n open paren 1 minus p close paren is greater than 5<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Method:<\/strong> The binomial distribution is approximated using a normal distribution with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03bc<\/mi><mo>=<\/mo><mi>n<\/mi><mi>p<\/mi><\/mrow><annotation encoding=\"text\/plain\">mu equals n p<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c3<\/mi><mo>=<\/mo><msqrt><mrow><mi>n<\/mi><mi>p<\/mi><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo>)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"text\/plain\">sigma equals the square root of n p open paren 1 minus p close paren end-root<\/annotation><\/semantics><\/math>.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">4. Key Concepts &amp; Skills&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Distinguish Variable Types:<\/strong> Identifying when to use discrete (binomial) vs. continuous (normal) distributions.<\/li>\n\n\n\n<li><strong>Calculate Probabilities:<\/strong> Finding probabilities for ranges (e.g., <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo>(<\/mo><mi>X<\/mi><mo>\u2264<\/mo><mi>a<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">cap P open paren cap X is less than or equal to a close paren<\/annotation><\/semantics><\/math> or <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo>(<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>X<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">cap P open paren a is less than cap X is less than b close paren<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Interpret Graphs:<\/strong> Recognizing features of the normal probability density function (PDF).<\/li>\n\n\n\n<li><strong>Statistical Inference:<\/strong> Using these distributions to interpret data, such as in interval estimates for proportions.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">These topics prepare students for Exam 2 (Technology Active) and form part of the Internal Assessment (IA3).&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">********************************************************************************************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u6570\u5b66\u6307\u5c0e\u6cd5\u30e6\u30cb\u30c3\u30c84\u3067\u306f\u3001\u4e8c\u9805\u5206\u5e03\u3068\u6b63\u898f\u5206\u5e03\u306f\u78ba\u7387\u3068\u7d71\u8a08\u306e\u4e2d\u6838\u3092\u6210\u3059\u8981\u7d20\u3067\u3042\u308a\u3001\u96e2\u6563\u30c7\u30fc\u30bf\u3068\u9023\u7d9a\u30c7\u30fc\u30bf\u306e\u30e2\u30c7\u30ea\u30f3\u30b0\u306b\u3088\u308b\u63a8\u8ad6\u306b\u91cd\u70b9\u3092\u7f6e\u3044\u3066\u3044\u307e\u3059\u3002\u30e6\u30cb\u30c3\u30c84\u306e\u30c8\u30d4\u30c3\u30af3\u3067\u306f\u3001\u901a\u5e38\u4e8c\u9805\u5206\u5e03\u306b\u3064\u3044\u3066\u3001\u30c8\u30d4\u30c3\u30af4\u3067\u306f\u9023\u7d9a\u78ba\u7387\u5909\u6570\u3068\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u6271\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u4e8c\u9805\u5206\u5e03\uff08\u96e2\u6563\uff09<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e8c\u9805\u5206\u5e03\u306f\u3001\u4e00\u5b9a\u56de\u6570\u306e\u72ec\u7acb\u3057\u305f\u300c\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u300d\u306b\u304a\u3051\u308b\u6210\u529f\u56de\u6570\u3092\u30e2\u30c7\u30eb\u5316\u3057\u307e\u3059\u3002\u5404\u8a66\u884c\u306e\u7d50\u679c\u306f\u6210\u529f\u304b\u5931\u6557\u306e2\u3064\u306e\u307f\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3b\u306a\u8981\u4ef6\uff1a\u8a66\u884c\u56de\u6570\uff08n\uff09\u3001\u5404\u8a66\u884c\u306b\u304a\u3051\u308b\u6210\u529f\u78ba\u7387\uff08p\uff09\u306f\u4e00\u5b9a\u3001\u8a66\u884c\u306f\u72ec\u7acb\u3067\u3042\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3b\u306a\u516c\u5f0f\uff1a<math data-latex=\"P(X=x)={n \\choose x}p^{x}(1-p)^{n-x}\"><semantics><mrow><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>=<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac linethickness=\"0px\"><mi>n<\/mi><mi>x<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><msup><mi>p<\/mi><mi>x<\/mi><\/msup><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mrow><mi>n<\/mi><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P(X=x)={n \\choose x}p^{x}(1-p)^{n-x}<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5e73\u5747\u3068\u5206\u6563\uff1a\u5e73\u5747 (<math data-latex=\"\u03bc\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"application\/x-tex\">\u03bc<\/annotation><\/semantics><\/math>) = <math data-latex=\"np\"><semantics><mrow><mi>n<\/mi><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">np<\/annotation><\/semantics><\/math>; \u5206\u6563 (<math data-latex=\"\u03c3^2\"><semantics><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><annotation encoding=\"application\/x-tex\">\u03c3^2<\/annotation><\/semantics><\/math>) = <math data-latex=\"np(1\u2212p)\"><semantics><mrow><mi>n<\/mi><mi>p<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">np(1\u2212p)<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5fdc\u7528\uff1a\u300c<math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> \u56de\u306e\u8a66\u884c\u3067\u3061\u3087\u3046\u3069 <math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math> \u56de\u6210\u529f\u3059\u308b\u78ba\u7387\u3092\u6c42\u3081\u308b\u300d\u3068\u3044\u3063\u305f\u554f\u984c\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>\u6b63\u898f\u5206\u5e03 (\u9023\u7d9a)<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b63\u898f\u5206\u5e03\uff08\u30ac\u30a6\u30b9\u5206\u5e03\uff09\u306f\u3001\u9023\u7d9a\u3057\u305f\u5bfe\u79f0\u7684\u306a\u300c\u30d9\u30eb\u578b\u300d\u66f2\u7dda\u3067\u3001\u8eab\u9577\u3001IQ \u30b9\u30b3\u30a2\u3001\u6e2c\u5b9a\u8aa4\u5dee\u306a\u3069\u306e\u9023\u7d9a\u30c7\u30fc\u30bf\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7279\u5fb4\uff1a\u5e73\u5747 (<math data-latex=\"\u03bc\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"application\/x-tex\">\u03bc<\/annotation><\/semantics><\/math>) \u3092\u4e2d\u5fc3\u306b\u5bfe\u79f0\u3067\u3001\u5e73\u5747\u3001\u4e2d\u592e\u5024\u3001\u6700\u983b\u5024\u304c\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u5e73\u5747 (<math data-latex=\"\u03bc\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"application\/x-tex\">\u03bc<\/annotation><\/semantics><\/math>) \u3068\u6a19\u6e96\u504f\u5dee (<math data-latex=\"\u03c3\"><semantics><mi>\u03c3<\/mi><annotation encoding=\"application\/x-tex\">\u03c3<\/annotation><\/semantics><\/math>) \u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7d4c\u9a13\u5247\uff0868-95-99.7%\u30eb\u30fc\u30eb\uff09\uff1a\u30c7\u30fc\u30bf\u306e\u7d0468%\u306f\u5e73\u5747\u5024\u306e1\u30b7\u30b0\u30de\u4ee5\u5185\u306b\u300195%\u306f2\u30b7\u30b0\u30de\u4ee5\u5185\u306b\u300199.7%\u306f3\u30b7\u30b0\u30de\u4ee5\u5185\u306b\u53ce\u307e\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6a19\u6e96\u6b63\u898f\u5206\u5e03\uff08\ud835\udc4d\uff09\uff1a<math data-latex=\"\u03bc\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"application\/x-tex\">\u03bc<\/annotation><\/semantics><\/math>=0\u3001\u30b7\u30b0\u30de\u304c1\u3068\u306a\u308b\u7279\u6b8a\u306a\u30b1\u30fc\u30b9\u3067\u3059\u3002\u5f0f <math data-latex=\" Z=\\frac{X-\\mu }{\\sigma }\"><semantics><mrow><mi>Z<\/mi><mo>=<\/mo><mfrac><mrow><mi>X<\/mi><mo>\u2212<\/mo><mi>\u03bc<\/mi><\/mrow><mi>\u03c3<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\"> Z=\\frac{X-\\mu }{\\sigma }<\/annotation><\/semantics><\/math>\uff08Z\u306f\u5206\u5b50\u304cX\u304b\u3089\u03bc\u3092\u5f15\u3044\u305f\u5206\u6570\u3001\u5206\u6bcd\u304c\u30b7\u30b0\u30de\u306e\u5206\u6570\uff09\u306f\u3001\u78ba\u7387\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30c6\u30af\u30ce\u30ed\u30b8\u30fc\u306e\u5229\u7528\uff1aCAS\u8a08\u7b97\u6a5f\u306f\u3001\u78ba\u7387\uff08\u4f8b\uff1anormCdf\uff09\u3068\u5206\u4f4d\u70b9\uff08\u9006\u6b63\u898f\u5206\u5e03\uff09\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li>\u4e8c\u9805\u5206\u5e03\u306e\u6b63\u898f\u8fd1\u4f3c<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u30e6\u30cb\u30c3\u30c84\u3067\u306f\u3001\u5927\u304d\u306a\u6a19\u672c\u30b5\u30a4\u30ba\uff08n\uff09\u306e\u5834\u5408\u3001\u4e8c\u9805\u5206\u5e03\u306f\u6b63\u898f\u5206\u5e03\u3067\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u3082\u5b66\u7fd2\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6761\u4ef6\uff1a\u901a\u5e38\u3001<math data-latex=\" np&gt;5 \"><semantics><mrow><mi>n<\/mi><mi>p<\/mi><mo>&gt;<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\"> np&gt;5 <\/annotation><\/semantics><\/math> and <math data-latex=\" n(1-p)&gt;5\"><semantics><mrow><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\"> n(1-p)&gt;5<\/annotation><\/semantics><\/math> \u3067\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u65b9\u6cd5\uff1a\u4e8c\u9805\u5206\u5e03\u306f\u3001<math data-latex=\"\u03bc\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"application\/x-tex\">\u03bc<\/annotation><\/semantics><\/math>  \u304c <math data-latex=\"n p\"><semantics><mrow><mi>n<\/mi><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n p<\/annotation><\/semantics><\/math> \u306b\u7b49\u3057\u304f\u3001<math data-latex=\"\\sigma =\\sqrt{np(1-p)}\"><semantics><mrow><mi>\u03c3<\/mi><mo>=<\/mo><msqrt><mrow><mi>n<\/mi><mi>p<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\sigma =\\sqrt{np(1-p)}<\/annotation><\/semantics><\/math>   \u306e\u6b63\u898f\u5206\u5e03\u3092\u4f7f\u7528\u3057\u3066\u8fd1\u4f3c\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<ol start=\"4\" class=\"wp-block-list\">\n<li>\u91cd\u8981\u306a\u6982\u5ff5\u3068\u30b9\u30ad\u30eb<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u5909\u6570\u306e\u7a2e\u985e\u306e\u533a\u5225\uff1a\u96e2\u6563\u5206\u5e03\uff08\u4e8c\u9805\u5206\u5e03\uff09\u3068\u9023\u7d9a\u5206\u5e03\uff08\u6b63\u898f\u5206\u5e03\uff09\u306e\u3069\u3061\u3089\u3092\u4f7f\u7528\u3059\u308b\u304b\u3092\u8b58\u5225\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u78ba\u7387\u306e\u8a08\u7b97\uff1a\u7bc4\u56f2\u306e\u78ba\u7387\u3092\u6c42\u3081\u308b\uff08\u4f8b\uff1a<math data-latex=\"P(X\u2264a)\"><semantics><mrow><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>\u2264<\/mo><mi>a<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P(X\u2264a)<\/annotation><\/semantics><\/math>\u3001\u307e\u305f\u306f <math data-latex=\"P(a&lt;X&lt;b)\"><semantics><mrow><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>X<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P(a&lt;X&lt;b)<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b0\u30e9\u30d5\u306e\u89e3\u91c8\uff1a\u6b63\u898f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff08PDF\uff09\u306e\u7279\u5fb4\u3092\u8a8d\u8b58\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7d71\u8a08\u7684\u63a8\u8ad6\uff1a\u3053\u308c\u3089\u306e\u5206\u5e03\u3092\u7528\u3044\u3066\u3001\u5272\u5408\u306e\u533a\u9593\u63a8\u5b9a\u306a\u3069\u306b\u304a\u3051\u308b\u30c7\u30fc\u30bf\u306e\u89e3\u91c8\u3092\u884c\u3046\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u308c\u3089\u306e\u30c8\u30d4\u30c3\u30af\u306f\u3001\u8a66\u9a132\uff08\u30c6\u30af\u30ce\u30ed\u30b8\u30fc\u30a2\u30af\u30c6\u30a3\u30d6\uff09\u306e\u6e96\u5099\u3068\u306a\u308a\u3001\u5185\u90e8\u8a55\u4fa1\uff08IA3\uff09\u306e\u4e00\u90e8\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In QLD Year 12 Mathematical Methods Unit 4, binomial and normal distributions are&nbsp;core components of probability and statistics, focusing on modeling discrete and continuous data to make inferences. Unit 4, Topic 3 typically covers the Binomial Distribution, while Topic 4 covers Continuous Random Variables and the Normal Distribution.&nbsp; 1. Binomial Distribution (Discrete)&nbsp; The binomial distribution [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1347","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1347","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1347"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1347\/revisions"}],"predecessor-version":[{"id":1454,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1347\/revisions\/1454"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1347"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1347"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1347"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}