4.9: Formula Review

4.1 Hypergeometric Distribution

\[
h(x)=\frac{\binom{A}{x}\binom{N-A}{n-x}}{\binom{N}{n}}
\]

4.2 Binomial Distribution

\(X \sim B(n, p)\) 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 \(x=0,1,2,3, \ldots, n\)
\(p=\) the probability of a success for any trial
\(q=\) the probability of a failure for any trial
\[
\begin{array}{l}
p+q=1 \\
q=1-p
\end{array}
\]

The mean of \(X\) is \(\mu=n p\). The standard deviation of \(X\) is \(\sigma=\sqrt{n p q}\).
\[
P(x)=\frac{n!}{x!(n-x)!} \cdot p^x q^{(n-x)}
\]
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

\[
P(X=x)=p(1-p)^{x-1}
\]
\(X \sim \mathrm{G}(\mathrm{p})\) 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 \(x=1,2,3, \ldots\)
\(p=\) 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 \(\mu=\frac{1}{p}\).
The standard deviation is \(\sigma=\sqrt{\frac{1-p}{p^2}}=\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}\).

4.4 Poisson Distribution

\(X \sim P(\mu)\) means that \(X\) has a Poisson probability distribution where \(X=\) the number of occurrences in the interval of interest.
\(X\) takes on the values \(x=0,1,2,3, \ldots\)
The mean \(\mu\) or \(\lambda\) is typically given.
The variance is \(\sigma^2=\mu\), and the standard deviation is
\[
\sigma=\sqrt{\mu}
\]

When \(P(\mu)\) is used to approximate a binomial distribution, \(\mu=n p\) where \(n\) represents the number of independent trials and \(p\) represents the probability of success in a single trial.
\[
P(x)=\frac{\mu^x e^{-\mu}}{x!}
\]