2.10 The Binomial Distribution

Justify why a random variable is or is not a binomial random variable.

• A binomial random variable, X, is a discrete random variable that counts the number of successes in repeated independent trials, n, that have only two possible outcomes (success or failure), with the probability of success p and the probability of failure 1−p .

Calculate the mean and standard deviation for a binomial distribution.

• If a random variable is binomial, its mean, µX, is np and its standard deviation, σX, is np(1-p .)

Interpret the mean, standard deviation, and probabilities for a binomial distribution.

• The mean, standard deviation, and probabilities for a binomial distribution should be interpreted in context.

Estimate probabilities of binomial random variables using data from a simulation.

• A probability distribution can be constructed using the rules of probability or estimated with a simulation. Probability, Random Variables, and Probability Distributions UNIT 2

Calculate probabilities for a binomial distribution.

• The probability that a binomial random variable, X, has exactly x successes for n independent trials, when the probability of success is p, is calculated as ( n) PX() | |= x = ppx - nx | x| ()1 - ( ) , x = 012,, ,, ... n. This is called the binomial probability function. Probability, Random Variables, and Probability Distributions UNIT 2 74