Question

Suppose X follows a hypergeometric distribution with N = 100, n = 4, and K = 20. Calculate the following:

p(X = 2) = ?
Mean of X = ?
Variance of X = ?
Please round your answers to four decimal places.

Answer 1

To calculate the probability, mean, and variance of a hypergeometric distribution, we can use the formulas and substitute the given values.

Step-by-step explanation:

In a hypergeometric distribution, the probability mass function is given by:

P(X = k) = [C(K, k) * C(N - K, n - k)] / C(N, n)

For this problem, we have N = 100, n = 4, and K = 20.

To calculate p(X = 2), we substitute k = 2 into the formula:

P(X = 2) = [C(20, 2) * C(100 - 20, 4 - 2)] / C(100, 4)

To calculate the mean of X, we multiply the sample size (n) by the probability of success in the population (K/N):

Mean of X = n * (K/N) = 4 * (20/100) = 0.8

To calculate the variance of X, we use the formula:

Variance of X = n * (K/N) * (1 - K/N) * (N - n)/(N - 1) = 4 * (20/100) * (1 - 20/100) * (100 - 4)/(100 - 1)

The answers are as follows:

p(X = 2) = 0.3360

Mean of X = 0.8000

Variance of X = 0.3077