Each egg laid by a hen falls onto the concrete floor of the

Each egg laid by a hen falls onto the concrete floor of the henhouse and cracks with probability p. If the number of eggs laid today by the hen has the Poisson distribution, with parameter ?, use moment generating function s to find the probability distribution of the number of uncracked eggs.

Solution

Let N denote the number of eggs laid. Then G N ( s ) = e ( s- 1) . Let X i , 1 , if the i th egg survives, , if the i th egg cracks. Thus, G X i ( s ) = (1- p ) s + p .

Since Z = N i =1 X i is the number of uncracked eggs, we have G Z ( s ) = G N ( G X i ( s )) = exp ( ((1- p ) s + p- 1) ) = e (1- p ) ( s- 1) . Therefore Z has the Poisson distribution with parameter (1- p )