2. Iron bars produced by a machine can either be classified as "Good" with probability p, or "Bad" with probability q = 1- p. The quality of each iron bar produced is independent of each other. (a) Suppose there are n iron bars produced by the machine. Write down the probability that there are exactly r "Good" iron bars among the n of them for all positive integers r. 4 (b) Suppose the machine keeps producing iron bars until k "Good" ones are obtained. Let X be the total number of iron bars produced. Show that the probability mass function of X is given by px(x) = (x - 1)^(1-p)²-*, x = k₁k+1,... 1 (c) In a single day, the machine produces exactly N = n iron bars, where N has mean and variance given by E(N) = a, Var(N) = 3. Let W be the total number of "Good" iron bars produced in a single day. i. Find E(W) and Var(W) in terms of p, a and 3. You can use the mean and variance of any standard distributions without proof as long as you state them clearly. In ii. Determine the covariance between W and N in terms of p, a, and 8. Simplify your answer as much as possible. 1-

2. Iron bars produced by a machine can either be classified as "Good" with probability p, or "Bad" with probability q = 1- p. The quality of each iron bar produced is independent of each other. (a) Suppose there are n iron bars produced by the machine. Write down the probability that there are exactly r "Good" iron bars among the n of them for all positive integers r. 4 (b) Suppose the machine keeps producing iron bars until k "Good" ones are obtained. Let X be the total number of iron bars produced. Show that the probability mass function of X is given by px(x) = (x - 1)^(1-p)²-*, x = k₁k+1,... 1 (c) In a single day, the machine produces exactly N = n iron bars, where N has mean and variance given by E(N) = a, Var(N) = 3. Let W be the total number of "Good" iron bars produced in a single day. i. Find E(W) and Var(W) in terms of p, a and 3. You can use the mean and variance of any standard distributions without proof as long as you state them clearly. In ii. Determine the covariance between W and N in terms of p, a, and 8. Simplify your answer as much as possible. 1-

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter12: Probability

Section12.CR: Chapter 12 Review

Problem 84CR

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