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**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 the Bernoulli probability function is

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**Mean and Variance of a Bernoulli Random Variable**

The mean is: And the variance is:

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**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.

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**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, , n

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**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, and variance,

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**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) = = 0.337

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**Binomial Probabilities, n = 100, p =0.40 (Figure 5.10)**

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