### 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable

The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows:

- Each probability is between zero and one, inclusive (
*inclusive*means to include zero and one). - The sum of the probabilities is one.

### 4.2 Mean or Expected Value and Standard Deviation

The expected value, or mean, of a discrete random variable predicts the long-term results of a statistical experiment that has been repeated many times. The standard deviation of a probability distribution is used to measure the variability of possible outcomes.

### 4.3 Binomial Distribution (Optional)

A statistical experiment can be classified as a binomial experiment if the following conditions are met:

- There are a fixed number of trials,
*n.* - There are only two possible outcomes, called
*success*and*failure,*for each trial; the letter*p*denotes the probability of a success on one trial and*q*denotes the probability of a failure on one trial. - The
*n*trials are independent and are repeated using identical conditions.

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable *X* = the number of successes obtained in the *n* independent trials. The mean of *X* can be calculated using the formula *μ* = *np*, and the standard deviation is given by the formula σ = $\text{}\sqrt{npq}$.

### 4.4 Geometric Distribution (Optional)

There are three characteristics of a geometric experiment.

- There are one or more Bernoulli trials with all failures except the last one, which is a success.
- In theory, the number of trials could go on forever; there must be at least one trial.
- The probability,
*p*, of a success and the probability,*q*, of a failure are the same for each trial.

In a geometric experiment, define the discrete random variable *X* as the number of independent trials until the first success. We say that *X* has a geometric distribution and write *X* ~ *G*(*p*) where *p* is the probability of success in a single trial.

The mean of the geometric distribution *X* ~ *G*(*p*) is *μ* = $\sqrt{\frac{\text{}\text{1}-p}{{p}^{2}}}$ = $\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}$.

### 4.5 Hypergeometric Distribution (Optional)

A hypergeometric experiment is a statistical experiment with the following properties:

- You take samples from two groups.
- You are concerned with a group of interest, called the first group.
- You sample without replacement from the combined groups.
- Each pick is not independent, since sampling is without replacement.
- You are not dealing with Bernoulli trials.

The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable *X* = the number of items from the group of interest. The distribution of *X* is denoted *X* ~ *H*(*r*, *b*, *n*), where *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. It follows that

*n*≤

*r*+

*b*. The mean of

*X*is

*μ*= $\frac{nr}{r\text{+}b}$ and the standard deviation is

*σ*= $\sqrt{\frac{rbn(r\text{+}b\text{\u2212}n)}{{(r\text{+}b)}^{2}\text{\hspace{0.17em}}(r\text{+}b-\text{1}\text{)}}}$.

### 4.6 Poisson Distribution (Optional)

A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a *fixed interval* of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is *small* (less than or equal to .05) and the number of trials is *large* (greater than or equal to 20).