4.1 Hypergeometric Distribution

h(x)=(Ax)(N−An−x)(Nn)

$ℎ\left(�\right)=\frac{\left(\begin{array}{c}�\\ �\end{array}\right)\left(\begin{array}{c}�–�\\ �–�\end{array}\right)}{\left(\begin{array}{c}�\\ �\end{array}\right)}$ 

4.2 Binomial Distribution

X ~ B(n, p) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p.

X = the number of successes in n independent trials

n = the number of independent trials

X takes on the values x = 0, 1, 2, 3, …, n

p = the probability of a success for any trial

q = the probability of a failure for any trial

p + q = 1

q = 1 – p

The mean of X is μ = np. The standard deviation of X is σ = npq−−−√

$\sqrt{���}$.

where P(X) is the probability of X successes in n trials when the probability of a success in ANY ONE TRIAL is p.

4.3 Geometric Distribution

P(X=x)=p(1−p)x−1

$�(�=�)=�(1-�{)}^{�-1}$ 

X ~ G(p) means that the discrete random variable X has a geometric probability distribution with probability of success in a single trial p.

X = the number of independent trials until the first success

X takes on the values x = 1, 2, 3, …

p = the probability of a success for any trial

q = the probability of a failure for any trial p + q = 1
q = 1 – p

The mean is μ = 1p

$\frac{1}{�}$.

The standard deviation is σ = 1 – pp2−−−−√

$\sqrt{\frac{1–�}{{�}^2}}$ = 1p(1p−1)−−−−−−−−√

$\sqrt{\frac{1}{�}\left(\frac{1}{�}-1\right)}$ .

4.4 Poisson Distribution

X ~ P(μ) means that X has a Poisson probability distribution where X = the number of occurrences in the interval of interest.

X takes on the values x = 0, 1, 2, 3, …

The mean μ or λ is typically given.

The variance is σ² = μ, and the standard deviation is
σ = μ−−√

$�\text{ = }\sqrt{�}$.

When P(μ) is used to approximate a binomial distribution, μ = np where n represents the number of independent trials and p represents the probability of success in a single trial.