Discrete Random Variable.ppt

Standard Deviation The positive square root of the variance is called the standard deviation (SD). * Example 5.15 The annual gross earnings (年收入) of a certain rock singer are a random variable with an expected value of $400,000 and a standard deviation of $80,000. The singer’s manager receives 15 percent of this amount. Determine the expected value and standard deviation of the amount received by the manager. Solution Let X denote the earnings (in units of $1000) of the singer, then the manager earns 0.15X. E[0.15X] = 0.15E[X] = 60 Var(0.15X) = (0.15)2Var(X) SD(0.15X) = 0.15 SD(X) = 12 The amount received by the manager is a random variable with an expected value of $60,000 and a standard deviation of $12,000. * Binomial Random Variables Suppose that n independent subexperiments (or trials) are performed, each of which results in either a “success” with probability p or a “failure” with probability 1 ? p. If X is the total number of successes that occur in n trials, then X is said to be a binomial random variable with parameters n and p. Suppose we are interested in the probability that 3 independent trials, each of which is a success with probability p, will result in a total of 2 successes. To determine this probability, consider all the outcomes that give rise to exactly 2 successes: (s, s, f ), (s, f, s), (f, s, s) The outcome (s, f, s) means that the first trial is a success, the second a failure, and the third a success. By the assumed independence of the trials, each of these outcomes has probability p2(1 ? p). The probability of a total of 2 successes in the 3 trials is 3p2(1 ? p). * Binomial Random Variables where For example, the probability that there are no successes in n trials is * Binomial Random Variables The probabilities of three binomial random variables with respective parameters n = 10, p = 0.5, n = 10, p = 0.3, and n = 10, p = 0.6 are presented in Fig. 5.3. * Example 5.16 Three fair coin