Đang chuẩn bị liên kết để tải về tài liệu:
Class Notes in Statistics and Econometrics Part 3

CHAPTER 5 Specific Random Variables. 5.1. Binomial We will begin with mean and variance of the binomial variable, i.e., the number of successes in n independent repetitions of a Bernoulli trial (3.7.1). The binomial variable has the two parameters n and p. Let us look first at the case n = 1 | CHAPTER 5 Specific Random Variables 5.1. Binomial We will begin with mean and variance of the binomial variable i.e. the number of successes in n independent repetitions of a Bernoulli trial 3.7.1 . The binomial variable has the two parameters n and p. Let us look first at the case n 1 in which the binomial variable is also called indicator variable If the event A has probability p then its complement A has the probability q 1 p. The indicator variable of A which assumes the value 1 if A occurs and 0 if it doesn t has expected value p and variance pq. For the binomial variable with n observations which is the sum of n independent indicator variables the expected value mean is np and the variance is npq. 139 140 5. SPECIFIC RANDOM VARIABLES PROBLEM 79. The random variable x assumes the value a with probability p and the value b with probability q 1 p. Show that var x pq a b 2. ANSWER. E x pa qb var x E x2 E x 2 pa2 qb2 pa qb 2 p p2 a2 2pqab q q2 b2 pq a b 2. For this last equality we need p p2 p 1 p pq. The Negative Binomial Variable is like the binomial variable derived from the Bernoulli experiment but one reverses the question. Instead of asking how many successes one gets in a given number of trials one asks how many trials one must make to get a given number of successes say r successes. First look at r 1. Let t denote the number of the trial at which the first success occurs. Then 5.1.1 Pr t n pqn-1 n 1 2 . . This is called the geometric probability. Is the probability derived in this way a-additive The sum of a geometrically declining sequence is easily computed 5.1.2 1 q q2 q3 s Now multiply by q 5.1.3 q q2 q3 qs Now subtract and write 1 q p 5.1.4 1 ps 5.1. BINOMIAL 141 Equation 5.1.4 means 1 p pq pq2 i.e. the sum of all probabilities is indeed 1. Now what is the expected value of a geometric variable Use definition of expected value of a discrete variable E t p 2 1 kqk-1. To evaluate the infinite sum solve 5.1.4 for s 5.1.5 s i or 1 q q2 q3 q4 V qk p k 0 1