{"id":2874,"date":"2024-10-24T17:00:03","date_gmt":"2024-10-24T15:00:03","guid":{"rendered":"https:\/\/ellistat.com\/?post_type=ressource-pedagogiqu&#038;p=2874"},"modified":"2024-10-24T17:01:21","modified_gmt":"2024-10-24T15:01:21","slug":"hypergeometric-law","status":"publish","type":"ressource-pedagogiqu","link":"https:\/\/ellistat.com\/en\/ressource-pedagogiqu\/loi-hypergeometrique\/","title":{"rendered":"Hypergeometric law"},"content":{"rendered":"<p>The hypergeometric law is a discrete probability distribution that models the number of successes in a fixed-size sample drawn without replacement from a finite population containing a specified number of success and failure objects. It is often used to model situations where selection is not independent, as each draw affects the composition of the population.&nbsp;<\/p>\n\n\n\n<p>Here are the key features of the hypergeometric law:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Finite population<\/strong> The total population contains a fixed number of objects, e.g. cards, marbles, coins, etc.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Objects of success and failure<\/strong> The population is divided into two groups: success objects (e.g. marked cards, winning marbles) and failure objects (e.g. unmarked cards, losing marbles).&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Pulling without replacement<\/strong> The sample is drawn without putting the objects back into the population after each draw. This means that the probability of drawing an object changes after each draw.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p>The probability mass function (PMF) of the hypergeometric law is given by the formula :&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\ud835\udc43(\ud835\udc4b=\ud835\udc58)=\ud835\udc36\ud835\udc3e\ud835\udc58\u2217\ud835\udc36\ud835\udc41-\ud835\udc3e\ud835\udc5b-\ud835\udc58\ud835\udc36\ud835\udc41\ud835\udc5bPX=k=CkK\u2217Cn-kN-KCnN<\/span><\/p>\n\n\n\n<p>where :&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>N is the total population size,\u00a0<\/li>\n\n\n\n<li>K is the number of successful objects in the population,\u00a0<\/li>\n\n\n\n<li>n is the sample size,\u00a0<\/li>\n\n\n\n<li>k is the number of successes in the sample,\u00a0<\/li>\n<\/ul>\n\n\n\n<p>The mean (expectation) of the hypergeometric distribution is :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\mu=n*\\frac{K}{N}<\/span><\/p>\n\n\n\n<p>The variance of a binomial distribution is :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma=n<em>\\frac{K<\/em>(N-K)*(N-n)}{N^{2}(N-1)}<\/span><\/p>\n\n\n\n<p>The hypergeometric law is used in fields such as statistics, genetics, economics and others where samples are drawn without replacement from a finite population. It differs from the binomial law in that it takes into account the evolution of the probability of success with each draw.&nbsp;<\/p>","protected":false},"featured_media":0,"template":"","meta":{"_acf_changed":false},"menu-ressource-pedagogique":[],"class_list":["post-2874","ressource-pedagogiqu","type-ressource-pedagogiqu","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Loi Hyperg\u00e9om\u00e9trique - Ellistat<\/title>\n<meta name=\"description\" content=\"La loi hyperg\u00e9om\u00e9trique est souvent utilis\u00e9e pour mod\u00e9liser des situations o\u00f9 la s\u00e9lection n&#039;est pas ind\u00e9pendante.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ellistat.com\/en\/educational-resource\/hypergeometric-law\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Loi Hyperg\u00e9om\u00e9trique - 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