Question

The probability mass function (pmf) for the Poisson distribution can be regarded as a limiting form of the binomial pmf if n o and p 0 with np = fi constant. (a) Suppose that 1% of all transistors produced by a certain company are defective. 100 of these chips are selected from the assembly line, Calculate the probability that exactly three of the chips are defective using both a binomial distribution and a Poisson distribution. In each case you may assume the assumptions are met but must clearly state the parameter/s of the distribution. (b) Suppose instead that 0.1% of the transistors are defective and 1000 chips are selected. Calculate the probability that exactly three of the chips are defective using both a binomial distribution and a Poisson distribution. In each case you may assume the Assumptions are met but must clearly state the parameters of the distribution. (c) Compare your answers to (a) and (b). What do you notice about using the Poisson distribution as a limiting form of the binomial distribution? (d) One property of the Poisson distribution is that mean and variance are the same. i. Calculate and compare the mean and variance of the Binomial random variables in (a) and (b) ii. Show that if X Binomial(n,p), then as n oo with np = fi constant lim Var(X) = E[X] P-0

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Answer 1

Answer:

1.7 @ 1.1. of productionarse transistor in the detective þ= 10%. = 0.01 6 paq Piaga" Number of chips selected (n =100 poinson parameter 1. M = up= loox oool = here the p = 0.1% =0.1= 0.001 a 9-15 - 0.999 Selected chips (4)=1000 exactly three chips defective P(x=3 Binomial) = 0.06198 poissou approximation = npr 1000x0.4 en 3.999 pe=s) = ép = 0.0613 poisson is a good approxinet on of Binomial

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