 Show
The hypergeometric distribution is discrete. It is similar to the binomial distribution. Both describe the number of times a particular event occurs in a fixed number of trials. However, binomial distribution trials are independent, while hypergeometric distribution trials change the success rate for each subsequent trial and are called “trials without replacement”. The hypergeometric distribution can be used for sampling problems such as the chance of picking a defective part from a box (without returning parts to the box for the next trial). Parameters Success, Trials, Population Conditions The hypergeometric distribution is used under these conditions:
The hypergeometric distribution is a discrete distribution that models the number of events in a fixed sample size when you know the total number of items in the population that the sample is from. Each item in the sample has two possible outcomes (either an event or a nonevent). The samples are without replacement, so every item in the sample is different. When an item is chosen from the population, it cannot be chosen again. Therefore, an item's chance of being selected increases on each trial, assuming that it has not yet been selected. Use the hypergeometric distribution for samples that are drawn from relatively small populations, without replacement. For example, the hypergeometric distribution is used in Fisher's exact test to test the difference between two proportions, and in acceptance sampling by attributes for sampling from an isolated lot of finite size. The hypergeometric distribution is defined by 3 parameters: population size, event count in population, and sample size. For example, you receive one special order shipment of 500 labels. Suppose that 2% of the labels are defective. The event count in the population is 10 (0.02 * 500). You sample 40 labels and want to determine the probability of 3 or more defective labels in that sample. The probability of 3 of more defective labels in the sample is 0.0384. Hypergeometric
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population of size that contains exactly objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of successes in draws with replacement. Definitions[edit]Probability mass function[edit]The following conditions characterize the hypergeometric distribution:
A random variable follows the hypergeometric distribution if its probability mass function (pmf) is given by[1] where The pmf is positive when . A random variable distributed hypergeometrically with parameters , and is written and has probability mass function above. Combinatorial identities[edit]As required, we have which essentially follows from Vandermonde's identity from combinatorics. Also note that This identity can be shown by expressing the binomial coefficients in terms of factorials and rearranging the latter, but it also follows from the symmetry of the problem. Indeed, consider two rounds of drawing without replacement. In the first round, out of neutral marbles are drawn from an urn without replacement and coloured green. Then the colored marbles are put back. In the second round, marbles are drawn without replacement and colored red. Then, the number of marbles with both colors on them (that is, the number of marbles that have been drawn twice) has the hypergeometric distribution. The symmetry in and stems from the fact that the two rounds are independent, and one could have started by drawing balls and colouring them red first. Properties[edit]Working example[edit]The classical application of the hypergeometric distribution is sampling without replacement. Think of an urn with two colors of marbles, red and green. Define drawing a green marble as a success and drawing a red marble as a failure (analogous to the binomial distribution). If the variable N describes the number of all marbles in the urn (see contingency table below) and K describes the number of green marbles, then N − K corresponds to the number of red marbles. In this example, X is the random variable whose outcome is k, the number of green marbles actually drawn in the experiment. This situation is illustrated by the following contingency table:
Now, assume (for example) that there are 5 green and 45 red marbles in the urn. Standing next to the urn, you close your eyes and draw 10 marbles without replacement. What is the probability that exactly 4 of the 10 are green? Note that although we are looking at success/failure, the data are not accurately modeled by the binomial distribution, because the probability of success on each trial is not the same, as the size of the remaining population changes as we remove each marble. This problem is summarized by the following contingency table:
The probability of drawing exactly k green marbles can be calculated by the formula Hence, in this example calculate Intuitively we would expect it to be even more unlikely that all 5 green marbles will be among the 10 drawn. As expected, the probability of drawing 5 green marbles is roughly 35 times less likely than that of drawing 4. Symmetries[edit]Swapping the roles of green and red marbles: Swapping the roles of drawn and not drawn marbles: Swapping the roles of green and drawn marbles: These symmetries generate the dihedral group . Order of draws[edit]The probability of drawing any set of green and red marbles (the hypergeometric distribution) depends only on the numbers of green and red marbles, not on the order in which they appear; i.e., it is an exchangeable distribution. As a result, the probability of drawing a green marble in the draw is[2] This is an ex ante probability—that is, it is based on not knowing the results of the previous draws. Tail bounds[edit]Let and . Then for we can derive the following bounds:[3] where is the Kullback-Leibler divergence and it is used that .[4] If n is larger than N/2, it can be useful to apply symmetry to "invert" the bounds, which give you the following: [4] [5] Statistical Inference[edit]Hypergeometric test[edit]The hypergeometric test uses the hypergeometric distribution to measure the statistical significance of having drawn a sample consisting of a specific number of successes (out of total draws) from a population of size containing successes. In a test for over-representation of successes in the sample, the hypergeometric p-value is calculated as the probability of randomly drawing or more successes from the population in total draws. In a test for under-representation, the p-value is the probability of randomly drawing or fewer successes. The test based on the hypergeometric distribution (hypergeometric test) is identical to the corresponding one-tailed version of Fisher's exact test.[6] Reciprocally, the p-value of a two-sided Fisher's exact test can be calculated as the sum of two appropriate hypergeometric tests (for more information see[7]). The test is often used to identify which sub-populations are over- or under-represented in a sample. This test has a wide range of applications. For example, a marketing group could use the test to understand their customer base by testing a set of known customers for over-representation of various demographic subgroups (e.g., women, people under 30). [edit]Let and . where is the standard normal distribution function
The following table describes four distributions related to the number of successes in a sequence of draws:
Multivariate hypergeometric distribution[edit]Multivariate hypergeometric distribution
The model of an urn with green and red marbles can be extended to the case where there are more than two colors of marbles. If there are Ki marbles of color i in the urn and you take n marbles at random without replacement, then the number of marbles of each color in the sample (k1, k2,..., kc) has the multivariate hypergeometric distribution. This has the same relationship to the multinomial distribution that the hypergeometric distribution has to the binomial distribution—the multinomial distribution is the "with-replacement" distribution and the multivariate hypergeometric is the "without-replacement" distribution. The properties of this distribution are given in the adjacent table,[8] where c is the number of different colors and is the total number of marbles. Example[edit]Suppose there are 5 black, 10 white, and 15 red marbles in an urn. If six marbles are chosen without replacement, the probability that exactly two of each color are chosen is Occurrence and applications[edit]Application to auditing elections[edit]Samples used for election audits and resulting chance of missing a problem Election audits typically test a sample of machine-counted precincts to see if recounts by hand or machine match the original counts. Mismatches result in either a report or a larger recount. The sampling rates are usually defined by law, not statistical design, so for a legally defined sample size n, what is the probability of missing a problem which is present in K precincts, such as a hack or bug? This is the probability that k = 0. Bugs are often obscure, and a hacker can minimize detection by affecting only a few precincts, which will still affect close elections, so a plausible scenario is for K to be on the order of 5% of N. Audits typically cover 1% to 10% of precincts (often 3%),[9][10][11] so they have a high chance of missing a problem. For example, if a problem is present in 5 of 100 precincts, a 3% sample has 86% probability that k = 0 so the problem would not be noticed, and only 14% probability of the problem appearing in the sample (positive k): The sample would need 45 precincts in order to have probability under 5% that k = 0 in the sample, and thus have probability over 95% of finding the problem: Application to Texas hold'em poker[edit]In
hold'em poker players make the best hand they can combining the two cards in their hand with the 5 cards (community cards) eventually turned up on the table. The deck has 52 and there are 13 of each suit. For this example assume a player has 2 clubs in the hand and there are 3 cards showing on the table, 2 of which are also clubs. The player would like to know the probability of one of the next 2 cards to be
shown being a club to complete the flush. There are 4 clubs showing so there are 9 clubs still unseen. There are 5 cards showing (2 in the hand and 3 on the table) so there are still unseen. The probability that one of the next two cards turned is a club can be calculated using hypergeometric with and . (about 31.64%) The probability that both of the next two cards turned are clubs can be calculated using hypergeometric with and . (about 3.33%) The probability that neither of the next two cards turned are clubs can be calculated using hypergeometric with and . (about 65.03%) See also[edit]
References[edit]Citations[edit]
Sources[edit]
External links[edit]
Which of the following is true about the hypergeometric distribution quizlet?Which of the following is true about the hypergeometric distribution? The trials are not independent and the probability of success may change from trial to trial.
Which of the following is the property of hypergeometric experiment?A hypergeometric experiment is a statistical experiment that has the following properties: A sample of size n is randomly selected without replacement from a population of N items. In the population, k items can be classified as successes, and N - k items can be classified as failures.
Which one of the following best describes a probability distribution?Which of the following statements best describes a probability distribution? A list of the outcomes of a random experiment and the probability of each outcome.
Which of the following best describes the expected value of a discrete random variable?Which of the following best describes the expected value of a discrete random variable? It is the geometric average of all possible outcomes.
|
Which of the following events is best described using a hypergeometric distribution
zusammenhängende Posts
Urheberrechte © © 2024 toptenid.com Inc.
