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.
$$P(x)=\frac{\mu^x e^{-\mu}}{x!}$$