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Mathematical Statistics with Resampling and R by Tim Hesterberg, Laura Chihara

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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 images Bern(p).

images

Now, let X1, X2, . . . , Xn 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 ...

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