Random Variables One of the Bet Catholic Colleges 随机变量的一个最好的天主教大学.pptVIP

Binomial random variable The random variable X whose probability mass function is given by is said to be a binomial random variable with parameters n and p. A binomial random variable gives the number of “successes” that occur when n independent trials, each of which result in a “success” with probability p, are performed. Example. Let X be the number of girls born to a family with 5 children. X is a binomial r. v. with n = 5, p = 0.5. Theorem. For a binomial r. v. X with parameters n and p, Expected value for a binomial random variable: parameters n, p E[X] = The last summation equals 1, by the binomial theorem. Make sure you can justify all the steps shown above. The maximum value of p(i) for a binomial r.v. Let [t] denote the greatest integer less than or equal t. Theorem. For a binomial random variable with parameters n and p, 0 p 1, the maximum value of the probability mass function p(i) occurs when Example. Let n = 10, p = 0.5. Then the maximum of the probability mass function occurs at [11(0.5)] = 5. Example. Let n = 11, p = 0.5. Then the maximum of the probability mass function occurs at [12(0.5)] = 6. By symmetry, the maximum also occurs at 11 – 6 = 5. Poisson random variable The random variable X whose probability mass function is given by is said to be a Poisson random variable with parameter , and A Poisson r. v. may be used as an approximation for a binomial r. v. with parameters (n, p) provided n is large and p is small enough that np has a moderate size (that is, np = for some constant ). The Poisson approximation for a binomial is generally good if p 0.1 and np 10. If np 10, use the normal approximation in Chapter 7 of the textbook. Example. The number of misprints on a page in a book is Poisson. Example of Poisson Random Variable Problem. Suppose that, on average, in every three pages of a book there is one typogr