This example shows how to compute and plot the cdf of a hypergeometric distribution. Previously, we developed a similarity measure utilizing the hypergeometric distribution and Fisher’s exact test [ 10 ]; this measure was restricted to two-class data, i.e., the comparison of binary images and data vectors. In the second case, the events are that sample item \(r\) is type \(i\) and that sample item \(s\) is type \(j\). This follows from the previous result and the definition of correlation. Let \(z = n - \sum_{j \in B} y_j\) and \(r = \sum_{i \in A} m_i\). number of observations. The probability that both events occur is \(\frac{m_i}{m} \frac{m_j}{m-1}\) while the individual probabilities are the same as in the first case. Then Someone told me to use the multinomial distribution but I think the hypergeometric distribution should be used and I don't understand the difference between multinomial and hypergeometric. \(\newcommand{\E}{\mathbb{E}}\) \(\newcommand{\bs}{\boldsymbol}\) This follows immediately, since \(Y_i\) has the hypergeometric distribution with parameters \(m\), \(m_i\), and \(n\). Multivariate Hypergeometric Distribution. Note that \(\sum_{i=1}^k Y_i = n\) so if we know the values of \(k - 1\) of the counting variables, we can find the value of the remaining counting variable. The types of the objects in the sample form a sequence of \(n\) multinomial trials with parameters \((m_1 / m, m_2 / m, \ldots, m_k / m)\). The multivariate hypergeometric distribution is preserved when the counting variables are combined. n[i] times. The conditional probability density function of the number of spades given that the hand has 3 hearts and 2 diamonds. Example 4.21 A candy dish contains 100 jelly beans and 80 gumdrops. The combinatorial proof is to consider the ordered sample, which is uniformly distributed on the set of permutations of size \(n\) from \(D\). 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. Suppose again that \(r\) and \(s\) are distinct elements of \(\{1, 2, \ldots, n\}\), and \(i\) and \(j\) are distinct elements of \(\{1, 2, \ldots, k\}\). Let \(W_j = \sum_{i \in A_j} Y_i\) and \(r_j = \sum_{i \in A_j} m_i\) for \(j \in \{1, 2, \ldots, l\}\). Now let \(I_{t i} = \bs{1}(X_t \in D_i)\), the indicator variable of the event that the \(t\)th object selected is type \(i\), for \(t \in \{1, 2, \ldots, n\}\) and \(i \in \{1, 2, \ldots, k\}\). As before we sample \(n\) objects without replacement, and \(W_i\) is the number of objects in the sample of the new type \(i\). Effectively, we are selecting a sample of size \(z\) from a population of size \(r\), with \(m_i\) objects of type \(i\) for each \(i \in A\). Hello, I’m trying to implement the Multivariate Hypergeometric distribution in PyMC3. \(\E(X) = \frac{13}{4}\), \(\var(X) = \frac{507}{272}\), \(\E(U) = \frac{13}{2}\), \(\var(U) = \frac{169}{272}\). The difference is the trials are done WITHOUT replacement. Hypergeometric Distribution Formula – Example #1. In a bridge hand, find each of the following: Let \(X\), \(Y\), and \(U\) denote the number of spades, hearts, and red cards, respectively, in the hand. We also say that \((Y_1, Y_2, \ldots, Y_{k-1})\) has this distribution (recall again that the values of any \(k - 1\) of the variables determines the value of the remaining variable). \[ \frac{1913496}{2598960} \approx 0.736 \]. The following results now follow immediately from the general theory of multinomial trials, although modifications of the arguments above could also be used. The Hypergeometric Distribution Basic Theory Dichotomous Populations. The probability that the sample contains at least 4 republicans, at least 3 democrats, and at least 2 independents. In the first case the events are that sample item \(r\) is type \(i\) and that sample item \(r\) is type \(j\). Arguments Suppose that we have a dichotomous population \(D\). Fisher's noncentral hypergeometric distribution The special case \(n = 5\) is the poker experiment and the special case \(n = 13\) is the bridge experiment. For the approximate multinomial distribution, we do not need to know \(m_i\) and \(m\) individually, but only in the ratio \(m_i / m\). Let the random variable X represent the number of faculty in the sample of size that have blood type O-negative. \[ \P(Y_1 = y_1, Y_2 = y_2, \ldots, Y_k = y_k) = \binom{n}{y_1, y_2, \ldots, y_k} \frac{m_1^{y_1} m_2^{y_2} \cdots m_k^{y_k}}{m^n}, \quad (y_1, y_2, \ldots, y_k) \in \N^k \text{ with } \sum_{i=1}^k y_i = n \], Comparing with our previous results, note that the means and correlations are the same, whether sampling with or without replacement. X = the number of diamonds selected. the length is taken to be the number required. Dear R Users, I employed the phyper() function to estimate the likelihood that the number of genes overlapping between 2 different lists of genes is due to chance. Does the multivariate hypergeometric distribution, for sampling without replacement from multiple objects, have a known form for the moment generating function? logical; if TRUE, probabilities p are given as log(p). The random variable X = the number of items from the group of interest. "Y^Cj = N, the bi-multivariate hypergeometric distribution is the distribution on nonnegative integer m x n matrices with row sums r and column sums c defined by Prob(^) = F[ r¡\ fT Cj\/(N\ IT ay!). A population of 100 voters consists of 40 republicans, 35 democrats and 25 independents. \cor\left(I_{r i}, I_{s j}\right) & = \frac{1}{m - 1} \sqrt{\frac{m_i}{m - m_i} \frac{m_j}{m - m_j}} Introduction Example of a multivariate hypergeometric distribution problem. The multivariate hypergeometric distribution is preserved when the counting variables are combined. \(\newcommand{\N}{\mathbb{N}}\) The ordinary hypergeometric distribution corresponds to \(k = 2\). In particular, \(I_{r i}\) and \(I_{r j}\) are negatively correlated while \(I_{r i}\) and \(I_{s j}\) are positively correlated. For distinct \(i, \, j \in \{1, 2, \ldots, k\}\). Add Multivariate Hypergeometric Distribution to scipy.stats. For more information on customizing the embed code, read Embedding Snippets. \[ \P(Y_1 = y_1, Y_2 = y_2, \ldots, Y_k = y_k) = \binom{n}{y_1, y_2, \ldots, y_k} \frac{m_1^{(y_1)} m_2^{(y_2)} \cdots m_k^{(y_k)}}{m^{(n)}}, \quad (y_1, y_2, \ldots, y_k) \in \N_k \text{ with } \sum_{i=1}^k y_i = n \]. The multivariate hypergeometric distribution is generalization of hypergeometric distribution. An analytic proof is possible, by starting with the first version or the second version of the joint PDF and summing over the unwanted variables. Hi all, in recent work with a colleague, the need came up for a multivariate hypergeometric sampler; I had a look in the numpy code and saw we have the bivariate version, but not the multivariate one. Application and example. If six marbles are chosen without replacement, the probability that exactly two of each color are chosen is The distribution of the balls that are not drawn is a complementary Wallenius' noncentral hypergeometric distribution. \((Y_1, Y_2, \ldots, Y_k)\) has the multinomial distribution with parameters \(n\) and \((m_1 / m, m_2, / m, \ldots, m_k / m)\): The covariance of each pair of variables in (a). In the fraction, there are \(n\) factors in the denominator and \(n\) in the numerator. Results from the hypergeometric distribution and the representation in terms of indicator variables are the main tools. Then A hypergeometric distribution can be used where you are sampling coloured balls from an urn without replacement. Probability mass function and random generation Description. A probabilistic argument is much better. Effectively, we now have a population of \(m\) objects with \(l\) types, and \(r_i\) is the number of objects of the new type \(i\). Both heads and … We will compute the mean, variance, covariance, and correlation of the counting variables. It is used for sampling without replacement Recall that if \(A\) and \(B\) are events, then \(\cov(A, B) = \P(A \cap B) - \P(A) \P(B)\). Now you want to find the … Details. Where k=sum (x) , N=sum (n) and k<=N . \(\P(X = x, Y = y, Z = z) = \frac{\binom{40}{x} \binom{35}{y} \binom{25}{z}}{\binom{100}{10}}\) for \(x, \; y, \; z \in \N\) with \(x + y + z = 10\), \(\E(X) = 4\), \(\E(Y) = 3.5\), \(\E(Z) = 2.5\), \(\var(X) = 2.1818\), \(\var(Y) = 2.0682\), \(\var(Z) = 1.7045\), \(\cov(X, Y) = -1.6346\), \(\cov(X, Z) = -0.9091\), \(\cov(Y, Z) = -0.7955\). m-length vector or m-column matrix Details If there are Ki type i object in the urn and we take n draws at random without replacement, then the numbers of type i objects in the sample (k1, k2, …, kc) has the multivariate hypergeometric distribution. It is shown that the entropy of this distribution is a Schur-concave function of the block-size parameters. However, a probabilistic proof is much better: \(Y_i\) is the number of type \(i\) objects in a sample of size \(n\) chosen at random (and without replacement) from a population of \(m\) objects, with \(m_i\) of type \(i\) and the remaining \(m - m_i\) not of this type. MAXIMUM LIKELIHOOD ESTIMATION OF A MULTIVARIATE HYPERGEOMETRIC DISTRIBUTION WALTER OBERHOFER and HEINZ KAUFMANN University of Regensburg, West Germany SUMMARY. My latest efforts so far run fine, but don’t seem to sample correctly. \(\P(X = x, Y = y, \mid Z = 4) = \frac{\binom{13}{x} \binom{13}{y} \binom{22}{9-x-y}}{\binom{48}{9}}\) for \(x, \; y \in \N\) with \(x + y \le 9\), \(\P(X = x \mid Y = 3, Z = 2) = \frac{\binom{13}{x} \binom{34}{8-x}}{\binom{47}{8}}\) for \(x \in \{0, 1, \ldots, 8\}\). Description The mean and variance of the number of red cards. Use the inclusion-exclusion rule to show that the probability that a poker hand is void in at least one suit is Recall that since the sampling is without replacement, the unordered sample is uniformly distributed over the combinations of size \(n\) chosen from \(D\). Now i want to try this with 3 lists of genes which phyper() does not appear to support. In this case, it seems reasonable that sampling without replacement is not too much different than sampling with replacement, and hence the multivariate hypergeometric distribution should be well approximated by the multinomial. hypergeometric distribution. Negative hypergeometric distribution describes number of balls x observed until drawing without replacement to obtain r white balls from the urn containing m white balls and n black balls, and is defined as . distributions sampling mgf hypergeometric multivariate-distribution I think we're sampling without replacement so we should use multivariate hypergeometric. The dichotomous model considered earlier is clearly a special case, with \(k = 2\). In contrast, the binomial distribution describes the probability of k {\displaystyle k} successes in n The classical application of the hypergeometric distribution is sampling without replacement.Think of an urn with two types of marbles, black ones and white ones.Define drawing a white marble as a success and drawing a black marble as a failure (analogous to the binomial distribution). That is, a population that consists of two types of objects, which we will refer to as type 1 and type 0. Gentle, J.E. Consider the second version of the hypergeometric probability density function. More generally, the marginal distribution of any subsequence of \( (Y_1, Y_2, \ldots, Y_n) \) is hypergeometric, with the appropriate parameters. This appears to work appropriately. The Hypergeometric Distribution is like the binomial distribution since there are TWO outcomes. In the card experiment, a hand that does not contain any cards of a particular suit is said to be void in that suit. Thus \(D = \bigcup_{i=1}^k D_i\) and \(m = \sum_{i=1}^k m_i\). The probability density funtion of \((Y_1, Y_2, \ldots, Y_k)\) is given by The denominator \(m^{(n)}\) is the number of ordered samples of size \(n\) chosen from \(D\). It is used for sampling without replacement k out of N marbles in m colors, where each of the colors appears n [i] times. As in the basic sampling model, we sample \(n\) objects at random from \(D\). \cor\left(I_{r i}, I_{r j}\right) & = -\sqrt{\frac{m_i}{m - m_i} \frac{m_j}{m - m_j}} \\ Suppose now that the sampling is with replacement, even though this is usually not realistic in applications. \cov\left(I_{r i}, I_{s j}\right) & = \frac{1}{m - 1} \frac{m_i}{m} \frac{m_j}{m} The multivariate hypergeometric distribution is generalization of hypergeometric distribution. For fixed \(n\), the multivariate hypergeometric probability density function with parameters \(m\), \((m_1, m_2, \ldots, m_k)\), and \(n\) converges to the multinomial probability density function with parameters \(n\) and \((p_1, p_2, \ldots, p_k)\). Usually it is clear from context which meaning is intended. The multivariate hypergeometric distribution is also preserved when some of the counting variables are observed. Some googling suggests i can utilize the Multivariate hypergeometric distribution to achieve this. To define the multivariate hypergeometric distribution in general, suppose you have a deck of size N containing c different types of cards. As in the basic sampling model, we start with a finite population \(D\) consisting of \(m\) objects. \[ \P(Y_1 = y_1, Y_2 = y_2, \ldots, Y_k = y_k) = \frac{\binom{m_1}{y_1} \binom{m_2}{y_2} \cdots \binom{m_k}{y_k}}{\binom{m}{n}}, \quad (y_1, y_2, \ldots, y_k) \in \N^k \text{ with } \sum_{i=1}^k y_i = n \], The binomial coefficient \(\binom{m_i}{y_i}\) is the number of unordered subsets of \(D_i\) (the type \(i\) objects) of size \(y_i\). Use the inclusion-exclusion rule to show that the probability that a bridge hand is void in at least one suit is MultivariateHypergeometricDistribution [ n, { m1, m2, …, m k }] represents a multivariate hypergeometric distribution with n draws without replacement from a collection containing m i objects of type i. Random number generation and Monte Carlo methods. 12 HYPERGEOMETRIC DISTRIBUTION Examples: 1. The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. Now let \(Y_i\) denote the number of type \(i\) objects in the sample, for \(i \in \{1, 2, \ldots, k\}\). Springer. In this paper, we propose a similarity measure with a probabilistic interpretation, utilizing the multivariate hypergeometric distribution and the Fisher-Freeman-Halton test. Specifically, suppose that \((A_1, A_2, \ldots, A_l)\) is a partition of the index set \(\{1, 2, \ldots, k\}\) into nonempty, disjoint subsets. Usage Usually it is clear In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k {\displaystyle k} successes in n {\displaystyle n} draws, without replacement, from a finite population of size N {\displaystyle N} that contains exactly K {\displaystyle K} objects with that feature, wherein each draw is either a success or a failure. For example, we could have an urn with balls of several different colors, or a population of voters who are either democrat, republican, or independent. Suppose that \(r\) and \(s\) are distinct elements of \(\{1, 2, \ldots, n\}\), and \(i\) and \(j\) are distinct elements of \(\{1, 2, \ldots, k\}\). successes of sample x x=0,1,2,.. x≦n 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. The probability mass function (pmf) of the distribution is given by: Where: N is the size of the population (the size of the deck for our case) m is how many successes are possible within the population (if you’re looking to draw lands, this would be the number of lands in the deck) n is the size of the sample (how many cards we’re drawing) k is how many successes we desire (if we’re looking to draw three lands, k=3) For the rest of this article, “pmf(x, n)”, will be the pmf of the scenario we  The covariance and correlation between the number of spades and the number of hearts. Run the simulation 1000 times and compute the relative frequency of the event that the hand is void in at least one suit. \(\newcommand{\R}{\mathbb{R}}\) Note that the marginal distribution of \(Y_i\) given above is a special case of grouping. The distribution of (Y1,Y2,...,Yk) is called the multivariate hypergeometric distribution with parameters m, (m1,m2,...,mk), and n. We also say that (Y1,Y2,...,Yk−1) has this distribution (recall again that the values of any k−1 of the variables determines the value of the remaining variable). If we group the factors to form a product of \(n\) fractions, then each fraction in group \(i\) converges to \(p_i\). Suppose that \(m_i\) depends on \(m\) and that \(m_i / m \to p_i\) as \(m \to \infty\) for \(i \in \{1, 2, \ldots, k\}\). There is also a simple algebraic proof, starting from the first version of probability density function above. Calculates the probability mass function and lower and upper cumulative distribution functions of the hypergeometric distribution. Once again, an analytic argument is possible using the definition of conditional probability and the appropriate joint distributions. \[ Y_i = \sum_{j=1}^n \bs{1}\left(X_j \in D_i\right) \]. In this section, we suppose in addition that each object is one of \(k\) types; that is, we have a multitype population. You have drawn 5 cards randomly without replacing any of the cards. The distribution of \((Y_1, Y_2, \ldots, Y_k)\) is called the multivariate hypergeometric distribution with parameters \(m\), \((m_1, m_2, \ldots, m_k)\), and \(n\). It is used for sampling without replacement \(k\) out of \(N\) marbles in \(m\) colors, where each of the colors appears \(n_i\) times. For example when flipping a coin each outcome (head or tail) has the same probability each time. Examples. An alternate form of the probability density function of \(Y_1, Y_2, \ldots, Y_k)\) is These events are disjoint, and the individual probabilities are \(\frac{m_i}{m}\) and \(\frac{m_j}{m}\). A univariate hypergeometric distribution can be used when there are two colours of balls in the urn, and a multivariate hypergeometric distribution can be used when there are more than two colours of balls. \((W_1, W_2, \ldots, W_l)\) has the multivariate hypergeometric distribution with parameters \(m\), \((r_1, r_2, \ldots, r_l)\), and \(n\). Combinations of the grouping result and the conditioning result can be used to compute any marginal or conditional distributions of the counting variables. k out of N marbles in m colors, where each of the colors appears \(\P(X = x, Y = y, Z = z) = \frac{\binom{13}{x} \binom{13}{y} \binom{13}{z}\binom{13}{13 - x - y - z}}{\binom{52}{13}}\) for \(x, \; y, \; z \in \N\) with \(x + y + z \le 13\), \(\P(X = x, Y = y) = \frac{\binom{13}{x} \binom{13}{y} \binom{26}{13-x-y}}{\binom{52}{13}}\) for \(x, \; y \in \N\) with \(x + y \le 13\), \(\P(X = x) = \frac{\binom{13}{x} \binom{39}{13-x}}{\binom{52}{13}}\) for \(x \in \{0, 1, \ldots 13\}\), \(\P(U = u, V = v) = \frac{\binom{26}{u} \binom{26}{v}}{\binom{52}{13}}\) for \(u, \; v \in \N\) with \(u + v = 13\). We assume initially that the sampling is without replacement, since this is the realistic case in most applications. See Also The number of spades and number of hearts. 1. Let \(X\), \(Y\) and \(Z\) denote the number of spades, hearts, and diamonds respectively, in the hand. The conditional distribution of \((Y_i: i \in A)\) given \(\left(Y_j = y_j: j \in B\right)\) is multivariate hypergeometric with parameters \(r\), \((m_i: i \in A)\), and \(z\). The mean and variance of the number of spades. Five cards are chosen from a well shuffled deck. \end{align}. We have two types: type \(i\) and not type \(i\). The number of red cards and the number of black cards. The number of spades, number of hearts, and number of diamonds. eg. \begin{align} \(\newcommand{\var}{\text{var}}\) She obtains a simple random sample of of the faculty. \(\newcommand{\cov}{\text{cov}}\) A random sample of 10 voters is chosen. Let \(X\), \(Y\), \(Z\), \(U\), and \(V\) denote the number of spades, hearts, diamonds, red cards, and black cards, respectively, in the hand. N=sum(n) and k<=N. Let Wj = ∑i ∈ AjYi and rj = ∑i ∈ Ajmi for j ∈ {1, 2, …, l} \end{align}. 2. (2006). EXAMPLE 2 Using the Hypergeometric Probability Distribution Problem: Suppose a researcher goes to a small college of 200 faculty, 12 of which have blood type O-negative. Thus the outcome of the experiment is \(\bs{X} = (X_1, X_2, \ldots, X_n)\) where \(X_i \in D\) is the \(i\)th object chosen. For \(i \in \{1, 2, \ldots, k\}\), \(Y_i\) has the hypergeometric distribution with parameters \(m\), \(m_i\), and \(n\) Let \(D_i\) denote the subset of all type \(i\) objects and let \(m_i = \#(D_i)\) for \(i \in \{1, 2, \ldots, k\}\). $\begingroup$ I don't know any Scheme (or Common Lisp for that matter), so that doesn't help much; also, the problem isn't that I can't calculate single variate hypergeometric probability distributions (which the example you gave is), the problem is with multiple variables (i.e. The multinomial coefficient on the right is the number of ways to partition the index set \(\{1, 2, \ldots, n\}\) into \(k\) groups where group \(i\) has \(y_i\) elements (these are the coordinates of the type \(i\) objects). \begin{align} \[ \frac{32427298180}{635013559600} \approx 0.051 \], \(\newcommand{\P}{\mathbb{P}}\) The multivariate hypergeometric distribution is generalization of Where k=sum(x), A multivariate version of Wallenius' distribution is used if there are more than two different colors. The variances and covariances are smaller when sampling without replacement, by a factor of the finite population correction factor \((m - n) / (m - 1)\). For example, we could have. hygecdf(x,M,K,N) computes the hypergeometric cdf at each of the values in x using the corresponding size of the population, M, number of items with the desired characteristic in the population, K, and number of samples drawn, N.Vector or matrix inputs for x, M, K, and N must all have the same size. The number of (ordered) ways to select the type \(i\) objects is \(m_i^{(y_i)}\). Compute the cdf of a hypergeometric distribution that draws 20 samples from a group of 1000 items, when the group contains 50 items of the desired type. However, this isn’t the only sort of question you could want to ask while constructing your deck or power setup. Suppose that we observe \(Y_j = y_j\) for \(j \in B\). Recall that if \(I\) is an indicator variable with parameter \(p\) then \(\var(I) = p (1 - p)\). Again, an analytic proof is possible, but a probabilistic proof is much better. of numbers of balls in m colors. Practically, it is a valuable result, since in many cases we do not know the population size exactly. The binomial coefficient \(\binom{m}{n}\) is the number of unordered samples of size \(n\) chosen from \(D\). The multivariate hypergeometric distribution has the following properties: ... 4.1 First example Apply this to an example from wiki: Suppose there are 5 black, 10 white, and 15 red marbles in an urn. Find each of the following: Recall that the general card experiment is to select \(n\) cards at random and without replacement from a standard deck of 52 cards. Let Say you have a deck of colored cards which has 30 cards out of which 12 are black and 18 are yellow. Specifically, there are K_1 cards of type 1, K_2 cards of type 2, and so on, up to K_c cards of type c. (The hypergeometric distribution is simply a special case with c=2 types of cards.) This has the same re­la­tion­ship to the multi­n­o­mial dis­tri­b­u­tionthat the hy­per­ge­o­met­ric dis­tri­b­u­tion has to the bi­no­mial dis­tri­b­u­tion—the multi­n­o­mial dis­tri­b­… We investigate the class of splitting distributions as the composition of a singular multivariate distribution and a univariate distribution. If length(n) > 1, The following exercise makes this observation precise. Specifically, suppose that \((A, B)\) is a partition of the index set \(\{1, 2, \ldots, k\}\) into nonempty, disjoint subsets. 2. As with any counting variable, we can express \(Y_i\) as a sum of indicator variables: For \(i \in \{1, 2, \ldots, k\}\) Specifically, suppose that (A1, A2, …, Al) is a partition of the index set {1, 2, …, k} into nonempty, disjoint subsets. Thus the result follows from the multiplication principle of combinatorics and the uniform distribution of the unordered sample. for the multivariate hypergeometric distribution. 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And 18 are yellow this follows from the group of interest that is, a population of voters... Suppose that we have two types of cards t seem to sample correctly the marginal of... Also a simple algebraic proof, starting from the multiplication principle of combinatorics and the number of red cards the! Event that the population size \ ( m\ ) is very large compared to sample! Of multinomial trials, although modifications of the faculty case, with (... C different types of objects, have a known form for the moment generating function from context which meaning intended. Dish contains 100 jelly beans and 80 gumdrops are black and 18 are yellow corresponds to \ Y_i\... Is very large compared to the multi­n­o­mial dis­tri­b­u­tionthat the hy­per­ge­o­met­ric dis­tri­b­u­tion has to the sample \! Not know the multivariate hypergeometric distribution examples size exactly are yellow singular multivariate distribution and the definition of probability., since in many cases we do not know the population size \ ( i\ ) k... Done without replacement so we should use multivariate hypergeometric distribution is like the binomial distribution since there are \ n\!

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