Bernoulli Distribution A Bernoulli distribution arises from a
Bernoulli Distribution A Bernoulli distribution arises from a random experiment which can give rise to just two possible outcomes. These outcomes are usually labeled as either “success” or “failure. ” If p denotes the probability of a success and the probability of a failure is (1 - p ), the Bernoulli probability function is
Mean and Variance of a Bernoulli Random Variable The mean is: And the variance is:
Sequences of x Successes in n Trials The number of sequences with x successes in n independent trials is: Where n! = n x (x – 1) x (n – 2) x. . . x 1 and 0! = 1.
Binomial Distribution Suppose that a random experiment can result in two possible mutually exclusive and collectively exhaustive outcomes, “success” and “failure, ” and that is the probability of a success resulting in a single trial. If n independent trials are carried out, the distribution of the resulting number of successes “x” is called the binomial distribution Its probability distribution function for the binomial random variable X = x is: P(x successes in n independent trials)= for x = 0, 1, 2. . . , n
Mean and Variance of a Binomial Probability Distribution Let X be the number of successes in n independent trials, each with probability of success . The x follows a binomial distribution with mean, mean and variance, variance
Binomial Probabilities - An Example – (Example 5. 7) An insurance broker, Shirley Ferguson, has five contracts, and she believes that for each contract, the probability of making a sale is 0. 40. What is the probability that she makes at most one sale? P(at most one sale) = P(X 1) = P(X = 0) + P(X = 1) = 0. 078 + 0. 259 = 0. 337
Binomial Probabilities, n = 100, p =0. 40 (Figure 5. 10)
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