# Chapter 4 Formula Review

## 4.1 Hypergeometric Distribution

$$h(x)=\frac{\left(\begin{array}{l}{A} \\ {x}\end{array}\right)\left(\begin{array}{l}{N-A} \\ {n-x}\end{array}\right)}{\left(\begin{array}{l}{N} \\ {n}\end{array}\right)}$$

## 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 $$x = 0, 1, 2, 3, ..., n$$

$$p =$$ 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 $$\mu = np$$. The standard deviation of $$X$$ is $$\sigma=\sqrt{n p q}$$.

$P(x)=\frac{n !}{x !(n-x) !} \cdot p^{x} q^{(n-x)}\nonumber$

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

$$P(X=x)=p(1-p)^{x-1}$$

$$X \sim G(p)$$ 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 $$\mu = \frac{1}{p}$$.

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 $$x = 0, 1, 2, 3, ...$$

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, $$\mu = np$$ where n represents the number of independent trials and $$p$$ represents the probability of success in a single trial.

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