5 Binomial Random Variable and Probability Distribution – 5.1 - Motivating Example and Definitions: 5.2 - Binomial Distribution and Random Variable n A binomial random variable, X, is defined to be the number of “successes” in independent trials where the P(success) = p is constant for each of trials. In the definition above notice the following conditions need to be satisfied for a binomial experiment: 1. There is a fixed number of trials carried out. n 2. The outcome of a given trial is either a “success” or “failure”. 3. The probability of success (p) remains constant from trial to trial. = q 4. The trials are independent, that is the outcome of a given trial is not affected by the outcome of any other trial. Binomial Probability Function The factorial expression in front counts the number of ways it is possible to obtain x successes in trials while the remainder of the formula gives the probability of getting a n particular sequence with and x n x “successes” – “failures”. Mean # of successes = Standard deviation of the # of successes  np The JMP spreadsheet does all the probability calculations for the following: P(X = x) = probability of obtaining x in n trials. “success” P(X < x) = probability of obtaining x or less in n trials. “successes” P(X > x) = probability of obtaining x or more in n trials. “successes”
5 – Binomial Random Variable and Probability Distribution 5.1 - Motivating Example and Definitions: 5.2 - Binomial Distribution and Random Variable A binomial random variable, X, is defined to be the number of “successes” in n independent trials where the P(success) = p is constant for each of trials. In the definition above notice the following conditions need to be satisfied for a binomial experiment: 1. There is a fixed number of n trials carried out. 2. The outcome of a given trial is either a “success” or “failure”. 3. The probability of success (p) remains constant from trial to trial. = q 4. The trials are independent, that is the outcome of a given trial is not affected by the outcome of any other trial. Binomial Probability Function The factorial expression in front counts the number of ways it is possible to obtain x successes in n trials while the remainder of the formula gives the probability of getting a particular sequence with x “successes” and n – x “failures”. Mean # of successes  = np Standard deviation of the # of successes The JMP spreadsheet does all the probability calculations for the following: P(X = x) = probability of obtaining x “success” in n trials. P(X < x) = probability of obtaining x or less “successes” in n trials. P(X > x) = probability of obtaining x or more “successes” in n trials.
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