SECTION 4.3 More Discrete Probability Distributions 219 Summary of Discrete Probability Distributions The table summarizes the discrete probability distributions discussed in this chapter. Distribution Summary Formulas Binomial Distribution A binomial experiment satisfies these conditions. 1. The experiment has a fixed number n of independent trials. 2. There are only two possible outcomes for each trial. Each outcome can be classified as a success or as a failure. 3. The probability of success p is the same for each trial. 4. The random variable x counts the number of successful trials. The parameters of a binomial distribution are n and p. n = the number of trials x = the number of successes in n trials p = probability of success in a single trial q = probability of failure in a single trial q = 1 - p The probability of exactly x successes in n trials is P1x2 = nCxp xqn-x = n! 1n - x2! x! pxqn-x. m = np s 2 = npq s = 1npq Geometric Distribution A geometric distribution is a discrete probability distribution of a random variable x that satisfies these conditions. 1. A trial is repeated until a success occurs. 2. The repeated trials are independent of each other. 3. The probability of success p is the same for each trial. 4. The random variable x represents the number of the trial in which the first success occurs. The parameter of a geometric distribution is p. x = the number of the trial in which the first success occurs p = probability of success in a single trial q = probability of failure in a single trial q = 1 - p The probability that the first success occurs on trial number x is P1x2 = pqx-1. Poisson Distribution The Poisson distribution is a discrete probability distribution of a random variable x that satisfies these conditions. 1. The experiment consists of counting the number of times x an event occurs over a specified interval of time, area, or volume. 2. The probability of the event occurring is the same for each interval. 3. The number of occurrences in one interval is independent of the number of occurrences in other intervals. The parameter of the Poisson distribution is m. x = the number of occurrences in the given interval m = the mean number of occurrences in a given interval unit The probability of exactly x occurrences in an interval is P1x2 = m xe-m x! .
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