Consider the manufacture of Grandmother’sFudge Nut Butter Cookies. Grandmother has noted that the number of nuts in acookie is a random variable with a Poisson mass function and that the average...

Consider the manufacture of Grandmother’sFudge Nut Butter Cookies. Grandmother has noted that the number of nuts in acookie is a random variable with a Poisson mass function and that the average number ofnuts per cookie is 1.5.
a. What is the numerical value of the probability of having at least one nut in a randomlyselected cookie?
b. Determine the numerical value of the variance of the number of nuts per cookie.
c. Determine the probability that a box of exactly M cookies contains exactly the expected value of the number of nuts for a box of N cookies. (M = 1, 2, · · · ;N = 1, 2, · · ·). Hint:semantics here are very important. You need to consider different cases of M and N,even and odd and realize that nuts are always whole. Just do some examples and then generalize.
d. What is the probability that a nut selected at random goes into a cookie containing exactly K nuts? Hint: a nut must go into a cookie and so this distribution can not contain zero nuts. Therefore, a suitable normalization of the original Poisson is needed.