### 4.2 Mean or Expected Value and Standard Deviation

Mean or Expected Value: $\mu =\underset{x\in X}{{\displaystyle \Sigma}}\phantom{\rule{.30em}{0ex}}xP(x)$

Standard Deviation: $\sigma =\sqrt{\underset{x\in X}{{{\displaystyle \Sigma}}^{\text{}}}{(x-\mu )}^{2}P(x)}$

### 4.3 Binomial Distribution (Optional)

*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* = $\sqrt{npq}$.

### 4.4 Geometric Distribution (Optional)

*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* = $\frac{1}{p}$.

The standard deviation is *σ* = $\sqrt{\frac{1\text{}\u2013\text{}p}{{p}^{2}}}$ = $\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}$ .

### 4.5 Hypergeometric Distribution (Optional)

*X* ~ *H*(*r*, *b*, *n*) means that the discrete random variable *X* has a hypergeometric probability distribution with *r* = the size of the group of interest (first group), *b* = the size of the second group, and *n* = the size of the chosen sample.

*X* = the number of items from the group of interest that are in the chosen sample, and *X* may take on the values *x* = 0, 1, . . . , up to the size of the group of interest. The minimum value for *X* may be larger than zero in some instances.

*n* ≤ *r* + *b*

The mean of *X* is given by the formula *μ* = $\frac{nr}{r\text{+}b}$ and the standard deviation is = $\sqrt{\frac{rbn(r\text{+}b-n)}{{(r\text{+}b)}^{2}(r\text{+}b-\text{1)}}}$.

### 4.6 Poisson Distribution (Optional)

*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* is typically given.

The variance is *σ*^{2} = *μ*, and the standard deviation is

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.