4 Each of 150 newly manufactured items is examined and the n

4. Each of 150 newly manufactured items is examined and the number of scratches per item is recorded, yielding the following data: Number of scratches 0 1 2 3 4 5 6 7 Observed frequency 19 36 42 30 13 7 2 1 Let X = the number of scratches on a randomly selected item, and assume that X has a Poisson distribution with parameter p. (a) Show that EX = Mu and Var(X) = Mu^2. (b) Find the method of moments estimator of p. for a random sample of size n from a Poisson Mu, show that this estimator is unbiased and that its standard error equal to Mu. (c) Show that the MLE of Mu is the same as the MOM, given above.

Solution

Mean of X = 316/150 = 2.1067

Var (X) =978/150 - 2.1067^2

= 6.52 - 4.4382 = 2.0818

As mean and var are almost equal we can fit Poisson distribution for this

Estimator for mu = Expected value of x

x	0	1	2	3	4	5	6	7
f	19	36	42	30	13	7	2	1	150
fx	0	36	84	90	52	35	12	7	316
fx^2	0	36	168	270	208	175	72	49	978