9 A manuscript is sent to a small business typing firm The f

9. A manuscript is sent to a small business typing firm. The firm has typists A, B, and C. For any work, it is equally likely that each typist will receive an assignment. The new business owner is interested in tracking errors for his firm and has collected some data analysis on the error yE 3770 Fall 2015 ISyE 3770 Statistics and Applications Homework #3 distribution and on each typist. If it is typed by A, then the number of errors made is a Poisson random variable with mean 5; if typed by B then the number of errors is a Poisson random variable with mean 3.5; and if typed by C, then it is a Poisson random variable with mean 1.4. Let X denote the number of errors in the typed manuscript. Write an expression for the probability distribution of X.

Solution

For a single Poisson distribution,

P(n) = u^n e^(-u) / n!

where u = mean, x = the number of errors.

As typists are equally likely to get an assignment, each has probability of 1/3. Hence, we multiply each of their Poisson distributions by 1/3, then add them:

P(X) = (1/3)[ (5^X)e^(-5)/X! + (3.5^X)e^(-3.5)/X! + (1.4^X)e^(-1.4)/X!] [ANSWER]