Problem 17BB

Poisson Approximation to Binomial Distribution An experiment consists of rolling a single die 12 times and the variable x is the number of times that the outcome is 6.

a. Can the Poisson distribution be used to find the probability that the outcome of 6 occurs exactly 3 times? Why or why not?

b. If the Poisson distribution is used, is the result OK? Why or why not?

Answer :

Step 1 :

Given, An experiment consists of rolling a single die 12 times and the variable x is the number of times that the outcome is 6.

- Where n = 12 and p = ⅙

With n = 12 and p = 0.166, the condition n100is not satisfied. So we cannot use poisson distribution as an approximation to the binomial.

b) Consider a random variable “x” follows poisson distribution with sample size “n” and parameter “”.

i,e.X P()

Probability mass function of poisson distribution is given by

P(x) = , x = 0,1,2,...,n.

Where, = np = 12(⅙) = 2

P(3) = = 0.180

Therefore the binomial distribution is

P(x) = (1 - p

P(3) = (1 - 0.166

= 0.197

The results are not ok.