I plant 10 tomato seeds in the Spring. Each seed has a becoming a fully-grown tomato plant. Write X for the number of fully grown plants that I see Each fully grown tomato plant produces a number of tomatoes which is Poisson dis- tributed with parameter 5, independent of all the other plants and of X. WriteY for the total number of tomatoes that I grow. i. (5 marks) State the distribution of X, including any parameters. Write down the probability mass function, mean and variance of _) , and deduce the value of E(X? ). ii. (5 marks) For any given r, find the distribution of Y X - I. and give the value of the function A(@) - E[Y X - ~] (Hint: you may assume that the sum of independent Poisson random variables is Poisson, but you should specify the parameters carefully). iii. (3 marks) Using the tower law, find the expected number of tomatoes that I grow, E(Y). iv. (5 marks) Give the value of the function B(x) - EY(Y - 1)|X - =], and hence find the expected value EY (Y - 1). Deduce the value of the variance of the number of tomatoes that I grow, Var (Y ). (Hint: if Z is Poisson with parameter ) then EZ(Z - 1) - A?)