# APPENDIX B

# PROBABILITY DISTRIBUTIONS

In this section, we gather results about some special probability distributions. Many of these are typically covered in a first probability course, so the presentations for them are brief. A document with information on computing probabilities and quantiles in R is provided at https://sites.google.com/site/ChiharaHesterberg.

## B.1 THE BERNOULLI AND BINOMIAL DISTRIBUTIONS

Consider a random experiment that has only two outcomes, tossing a coin and recording “heads” or “tails.” This experiment is called a *Bernoulli trial*. We define a random variable *X* on the set of these two outcomes, letting *X* have the value 0 with probability *p* for one outcome, and value 1 with probability 1 − *p* for the other outcome. Then *X* is called a *Bernoulli* random variable. For instance, if we roll a die and consider seeing a 1 a “success” and anything else a “failure,” we can let *X* = 1 for a success with probability *p* = 1/6 and *X* = 0 for a failure with probability 1 − 1/6 = 5/6.

We denote a random variable *X* with probability *p* of success by *X* Bern(*p*).

Now, let *X*_{1}, *X*_{2}, . . . , *X*_{n} be *n* independent Bernoulli random variables, each with probability *p* of success, and let *Y* denote the number of successes. Then *Y* is a *binomial* random variable denoted by *Y* Binom(*n, p*) with probability ...