{"id":1345,"date":"2026-01-27T15:06:42","date_gmt":"2026-01-27T05:06:42","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1345"},"modified":"2026-01-28T19:45:39","modified_gmt":"2026-01-28T09:45:39","slug":"year12-math-3-4-1-discrete-and-continuous-random-variables","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1345","title":{"rendered":"Year12 MATH 3-4-1 Discrete and continuous random variables"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In QLD Year 12 Mathematical Methods Unit 4, <mark>the study of random variables focuses on quantifying outcomes of random phenomena, divided into discrete (counted) and continuous (measured) types<\/mark>. Unit 4 specifically delves into continuous random variables, the Normal distribution, and statistical inference, building upon previous studies of probability and discrete distributions.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>1. Discrete Random Variables (DRV) &#8211; <em>Review\/Context<\/em>&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">While often studied in earlier units, Discrete Random Variables (DRV) are essential for understanding the context of probability distributions.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition:<\/strong> Variables that take on distinct, countable, often integer values.<\/li>\n\n\n\n<li><strong>Examples:<\/strong> Number of students in a class, number of heads in coin tosses, or number of defective items in a sample.<\/li>\n\n\n\n<li><strong>Probability Distributions:<\/strong> Represented by tables or probability mass functions where the sum of probabilities equals 1.<\/li>\n\n\n\n<li><strong>Context in Unit 4:<\/strong> These are used to understand the transition to binomial distributions and provide a baseline for understanding random behavior.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2. Continuous Random Variables (CRV) &#8211; <em>Unit 4 Focus<\/em>&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Continuous Random Variables (CRV) represent data that can take on any value within an unbroken interval or range.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition:<\/strong> Measured quantities rather than counted, such as time, weight, height, or distance.<\/li>\n\n\n\n<li><strong>Key Concept:<\/strong> The probability of a CRV taking an exact, specific value is zero. Probabilities are calculated over intervals, not at specific points.<\/li>\n\n\n\n<li><strong>Probability Density Function (PDF):<\/strong> The probability of a range of values is found by calculating the area under a curve, defined by a function <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">f of x<\/annotation><\/semantics><\/math>.\n<ul class=\"wp-block-list\">\n<li><strong>Requirements:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>\u2265<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">f of x is greater than or equal to 0<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mo largeop=\"true\">\u222b<\/mo><mrow><mo>\u2212<\/mo><mo>\u221e<\/mo><\/mrow><mo>\u221e<\/mo><\/msubsup><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">integral from negative infinity to infinity of f of x d x equals 1<\/annotation><\/semantics><\/math>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Cumulative Distribution Function (CDF):<\/strong> Defined as <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>F<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>=<\/mo><mi>P<\/mi><mo>(<\/mo><mi>X<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">cap F open paren x close paren equals cap P open paren cap X is less than or equal to x close paren<\/annotation><\/semantics><\/math>, which represents the accumulated probability up to a certain value.<\/li>\n\n\n\n<li><strong>Key Techniques:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Using integration to find probabilities, mean (expected value), variance, and standard deviation.<\/li>\n\n\n\n<li>Using relative frequencies and histograms from data to estimate probabilities.&nbsp;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>3. The Normal Distribution&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A major part of Unit 4 involves studying the normal distribution as a specific type of continuous random variable.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Normal Distribution Properties:<\/strong> Symmetric, bell-shaped curve defined by 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>Standardised Normal Distribution (\ud835\udc67-scores):<\/strong> Converting a normal variable <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>X<\/mi><annotation encoding=\"text\/plain\">cap X<\/annotation><\/semantics><\/math> to the standard normal variable <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>Z<\/mi><annotation encoding=\"text\/plain\">cap Z<\/annotation><\/semantics><\/math> (mean 0, variance 1) using <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\">z equals the fraction with numerator x minus mu and denominator sigma end-fraction<\/annotation><\/semantics><\/math> to compare samples and calculate probabilities.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>4. Key Differences&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><th>Feature&nbsp;<\/th><th>Discrete Random Variable<\/th><th>Continuous Random Variable<\/th><\/tr><tr><td><strong>Data Type<\/strong><\/td><td>Counted (e.g., 1, 2, 3)<\/td><td>Measured (e.g., 1.5, 2.75)<\/td><\/tr><tr><td><strong>Values<\/strong><\/td><td>Isolated\/Separated<\/td><td>Unbroken Interval<\/td><\/tr><tr><td><strong>Probability<\/strong><\/td><td><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><\/mrow><annotation encoding=\"text\/plain\">cap P open paren cap X equals x close paren<\/annotation><\/semantics><\/math> exists<\/td><td><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><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">cap P open paren cap X equals x close paren equals 0<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td><strong>Formula<\/strong><\/td><td>Summation (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mo largeop=\"true\" movablelimits=\"true\">\u2211<\/mo><annotation encoding=\"text\/plain\">sum of<\/annotation><\/semantics><\/math>)<\/td><td>Integration (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mo largeop=\"true\">\u222b<\/mo><annotation encoding=\"text\/plain\">integral of<\/annotation><\/semantics><\/math>)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Summary of Unit 4 Learning Outcomes&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Students in QLD Mathematical Methods are required to:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Use PDFs and CDFs to calculate probabilities for continuous variables.<\/li>\n\n\n\n<li>Calculate the expected value (mean), variance, and standard deviation of a CRV.<\/li>\n\n\n\n<li>Apply normal distribution properties to solve problems.<\/li>\n\n\n\n<li>Use intervals of data to determine probabilities.&nbsp;<\/li>\n<\/ul>\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\u78ba\u7387\u5909\u6570\u306e\u5b66\u7fd2\u306b\u304a\u3044\u3066\u3001\u30e9\u30f3\u30c0\u30e0\u73fe\u8c61\u306e\u7d50\u679c\u3092\u5b9a\u91cf\u5316\u3059\u308b\u3053\u3068\u306b\u7126\u70b9\u3092\u5f53\u3066\u3001\u96e2\u6563\u578b\uff08\u30ab\u30a6\u30f3\u30c8\u578b\uff09\u3068\u9023\u7d9a\u578b\uff08\u6e2c\u5b9a\u578b\uff09\u306b\u5206\u3051\u3089\u308c\u307e\u3059\u3002\u30e6\u30cb\u30c3\u30c84\u3067\u306f\u3001\u7279\u306b\u9023\u7d9a\u578b\u78ba\u7387\u5909\u6570\u3001\u6b63\u898f\u5206\u5e03\u3001\u305d\u3057\u3066\u7d71\u8a08\u7684\u63a8\u8ad6\u306b\u3064\u3044\u3066\u6df1\u304f\u6398\u308a\u4e0b\u3052\u3001\u78ba\u7387\u3068\u96e2\u6563\u5206\u5e03\u306b\u95a2\u3059\u308b\u3053\u308c\u307e\u3067\u306e\u5b66\u7fd2\u3092\u57fa\u76e4\u3068\u3057\u3066\u5b66\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u96e2\u6563\u78ba\u7387\u5909\u6570\uff08DRV\uff09 &#8211; \u5fa9\u7fd2\/\u80cc\u666f<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u96e2\u6563\u78ba\u7387\u5909\u6570\uff08DRV\uff09\u306f\u3001\u3053\u308c\u307e\u3067\u306e\u30e6\u30cb\u30c3\u30c8\u3067\u3082\u983b\u7e41\u306b\u5b66\u7fd2\u3055\u308c\u3066\u3044\u307e\u3059\u304c\u3001\u78ba\u7387\u5206\u5e03\u306e\u6587\u8108\u3092\u7406\u89e3\u3059\u308b\u4e0a\u3067\u4e0d\u53ef\u6b20\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5b9a\u7fa9\uff1a\u7570\u306a\u308b\u3001\u53ef\u7b97\u306a\u3001\u591a\u304f\u306e\u5834\u5408\u6574\u6570\u3068\u306a\u308b\u5024\u3092\u3068\u308b\u5909\u6570\u3002<br>\u4f8b\uff1a\u30af\u30e9\u30b9\u306e\u751f\u5f92\u6570\u3001\u30b3\u30a4\u30f3\u6295\u3052\u3067\u8868\u304c\u51fa\u305f\u6570\u3001\u30b5\u30f3\u30d7\u30eb\u306b\u542b\u307e\u308c\u308b\u4e0d\u826f\u54c1\u306e\u6570\u306a\u3069\u3002<br>\u78ba\u7387\u5206\u5e03\uff1a\u78ba\u7387\u306e\u7dcf\u548c\u304c1\u3068\u306a\u308b\u8868\u307e\u305f\u306f\u78ba\u7387\u8cea\u91cf\u95a2\u6570\u3067\u8868\u3055\u308c\u307e\u3059\u3002<br>\u30e6\u30cb\u30c3\u30c84\u3067\u306e\u80cc\u666f\uff1a\u4e8c\u9805\u5206\u5e03\u3078\u306e\u79fb\u884c\u3092\u7406\u89e3\u3057\u3001\u30e9\u30f3\u30c0\u30e0\u306a\u632f\u308b\u821e\u3044\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u306e\u57fa\u6e96\u3092\u63d0\u4f9b\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>\u9023\u7d9a\u78ba\u7387\u5909\u6570\uff08CRV\uff09 &#8211; \u30e6\u30cb\u30c3\u30c84\u306e\u7126\u70b9<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u9023\u7d9a\u78ba\u7387\u5909\u6570\uff08CRV\uff09\u306f\u3001\u9014\u5207\u308c\u308b\u3053\u3068\u306e\u306a\u3044\u533a\u9593\u307e\u305f\u306f\u7bc4\u56f2\u5185\u3067\u4efb\u610f\u306e\u5024\u3092\u53d6\u308b\u53ef\u80fd\u6027\u306e\u3042\u308b\u30c7\u30fc\u30bf\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5b9a\u7fa9\uff1a\u6642\u9593\u3001\u4f53\u91cd\u3001\u8eab\u9577\u3001\u8ddd\u96e2\u306a\u3069\u3001\u6570\u3048\u3089\u308c\u308b\u306e\u3067\u306f\u306a\u304f\u6e2c\u5b9a\u3055\u308c\u308b\u91cf\u3002<br>\u4e3b\u8981\u6982\u5ff5\uff1aCRV\u304c\u7279\u5b9a\u306e\u5024\u3092\u6b63\u78ba\u306b\u53d6\u308b\u78ba\u7387\u306f0\u3067\u3059\u3002\u78ba\u7387\u306f\u7279\u5b9a\u306e\u70b9\u3067\u306f\u306a\u304f\u3001\u533a\u9593\u5168\u4f53\u3067\u8a08\u7b97\u3055\u308c\u307e\u3059\u3002<br>\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\uff08PDF\uff09\uff1a\u3042\u308b\u7bc4\u56f2\u306e\u5024\u306e\u78ba\u7387\u306f\u3001\u95a2\u6570 <math data-latex=\"f(x)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math> \u3067\u5b9a\u7fa9\u3055\u308c\u308b\u66f2\u7dda\u306e\u4e0b\u306e\u9762\u7a4d\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u3067\u6c42\u3081\u3089\u308c\u307e\u3059\u3002<br>\u8981\u4ef6: (<math data-latex=\"f(x)\\ge 0\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x)\\ge 0<\/annotation><\/semantics><\/math>) \u304b\u3064 (<math data-latex=\"\\int _{-\\infty }^{\\infty }f(x)dx=1\"><semantics><mrow><msubsup><mo movablelimits=\"false\">\u222b<\/mo><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mi>\u221e<\/mi><\/msubsup><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\int _{-\\infty }^{\\infty }f(x)dx=1<\/annotation><\/semantics><\/math>)<br>\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570 (CDF): <math data-latex=\"F(x)=P(X\\le x)\"><semantics><mrow><mi>F<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>P<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">F(x)=P(X\\le x)<\/annotation><\/semantics><\/math> \u3068\u5b9a\u7fa9\u3055\u308c\u3001\u3042\u308b\u5024\u307e\u3067\u306e\u7d2f\u7a4d\u78ba\u7387\u3092\u8868\u3057\u307e\u3059\u3002<br>\u4e3b\u8981\u306a\u624b\u6cd5: \u7a4d\u5206\u3092\u7528\u3044\u3066\u78ba\u7387\u3001\u5e73\u5747\uff08\u671f\u5f85\u5024\uff09\u3001\u5206\u6563\u3001\u6a19\u6e96\u504f\u5dee\u3092\u6c42\u3081\u308b\u3002\u30c7\u30fc\u30bf\u306e\u76f8\u5bfe\u5ea6\u6570\u3068\u30d2\u30b9\u30c8\u30b0\u30e9\u30e0\u3092\u7528\u3044\u3066\u78ba\u7387\u3092\u63a8\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li>\u6b63\u898f\u5206\u5e03<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u30e6\u30cb\u30c3\u30c84\u3067\u306f\u3001\u4e3b\u306b\u9023\u7d9a\u78ba\u7387\u5909\u6570\u306e\u4e00\u7a2e\u3067\u3042\u308b\u6b63\u898f\u5206\u5e03\u306b\u3064\u3044\u3066\u5b66\u7fd2\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b63\u898f\u5206\u5e03\u306e\u7279\u6027: \u5e73\u5747 (<math data-latex=\"\\mu\"><semantics><mi>\u03bc<\/mi><annotation encoding=\"application\/x-tex\">\\mu<\/annotation><\/semantics><\/math> ) \u3068\u6a19\u6e96\u504f\u5dee (<math data-latex=\"\\sigma\"><semantics><mi>\u03c3<\/mi><annotation encoding=\"application\/x-tex\">\\sigma<\/annotation><\/semantics><\/math> ) \u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u5bfe\u79f0\u7684\u306a\u30d9\u30eb\u578b\u66f2\u7dda\u3002<br>\u6a19\u6e96\u5316\u6b63\u898f\u5206\u5e03 (z \u30b9\u30b3\u30a2): (<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>) \u3092\u4f7f\u7528\u3057\u3066\u6b63\u898f\u5909\u6570 (<math data-latex=\"X\"><semantics><mi>X<\/mi><annotation encoding=\"application\/x-tex\">X<\/annotation><\/semantics><\/math>) \u3092\u6a19\u6e96\u6b63\u898f\u5909\u6570 (<math data-latex=\"Z\"><semantics><mi>Z<\/mi><annotation encoding=\"application\/x-tex\">Z<\/annotation><\/semantics><\/math>) (\u5e73\u5747 0\u3001\u5206\u6563 1) \u306b\u5909\u63db\u3057\u3001\u30b5\u30f3\u30d7\u30eb\u3092\u6bd4\u8f03\u3057\u3066\u78ba\u7387\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<ol start=\"4\" class=\"wp-block-list\">\n<li>\u4e3b\u306a\u9055\u3044<\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>\u7279\u5fb4<\/td><td>\u96e2\u6563\u78ba\u7387\u5909\u6570<\/td><td>\u9023\u7d9a\u78ba\u7387\u5909\u6570<\/td><\/tr><tr><td>\u30c7\u30fc\u30bf\u30bf\u30a4\u30d7<\/td><td>\u500b\u6570\u578b\uff08\u4f8b\uff1a1\u30012\u30013\uff09<\/td><td>\u6e2c\u5b9a\u5024\u578b\uff08\u4f8b\uff1a1.5\u30012.75\uff09<\/td><\/tr><tr><td>\u5024<\/td><td>\u5206\u96e2\u578b<\/td><td>\u9023\u7d9a\u533a\u9593<\/td><\/tr><tr><td>\u78ba\u7387<\/td><td>\ud835\udc43(\ud835\udc4b=\ud835\udc65)\u3000\u5b58\u5728<\/td><td><math data-latex=\"P(X=x)=0\"><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><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">P(X=x)=0<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td>\u8a08\u7b97\u5f0f<\/td><td>\u3000\u548c\uff08<math data-latex=\"\u2211\"><semantics><mo movablelimits=\"false\" lspace=\"0em\" rspace=\"0em\">\u2211<\/mo><annotation encoding=\"application\/x-tex\">\u2211<\/annotation><\/semantics><\/math>\uff09<\/td><td>\u7a4d\u5206\uff08<math data-latex=\"\u222b\"><semantics><mo movablelimits=\"false\" lspace=\"0em\" rspace=\"0em\">\u222b<\/mo><annotation encoding=\"application\/x-tex\">\u222b<\/annotation><\/semantics><\/math>\uff09<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u30e6\u30cb\u30c3\u30c84 \u5b66\u7fd2\u6210\u679c\u306e\u307e\u3068\u3081<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD\u6570\u5b66\u7684\u624b\u6cd5\u3092\u5b66\u3076\u751f\u5f92\u306f\u3001\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u6c42\u3081\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9023\u7d9a\u5909\u6570\u306e\u78ba\u7387\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570(PDF)\u3068\u7d2f\u7a4d\u5206\u5e03\u95a2\u6570(CDF) \u3092\u4f7f\u7528\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9023\u7d9a\u78ba\u7387\u5909\u6570(CRV) \u306e\u671f\u5f85\u5024\uff08\u5e73\u5747\uff09\u3001\u5206\u6563\u3001\u6a19\u6e96\u504f\u5dee\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b63\u898f\u5206\u5e03\u306e\u6027\u8cea\u3092\u9069\u7528\u3057\u3066\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30c7\u30fc\u30bf\u306e\u533a\u9593\u3092\u7528\u3044\u3066\u78ba\u7387\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In QLD Year 12 Mathematical Methods Unit 4, the study of random variables focuses on quantifying outcomes of random phenomena, divided into discrete (counted) and continuous (measured) types. Unit 4 specifically delves into continuous random variables, the Normal distribution, and statistical inference, building upon previous studies of probability and discrete distributions.&nbsp; 1. Discrete Random Variables [&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-1345","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1345","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=1345"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1345\/revisions"}],"predecessor-version":[{"id":1452,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1345\/revisions\/1452"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1345"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1345"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}