Statistics

Plz answer this question

2. Again, the chickenegg problem! A chicken lays a number of eggs, N, which follows a Poisson()\\) distribution. Each egg hatches a chick with probability 10, independently. Let X be the number of eggs which hatch, so X | N = n N Bin(n, p) and let Y be the number of eggs which don't hatch, so Y | N = n ~ Bin(n, 1 39). We also note that N = X + Y. (a) (2 pts) Find the marginal distribution of Y. Hint: Start from the joint PMF of Y and N, as shown for the marginal distribution of X during the class. (b) (2 pts) By the law of total probability, the following is the case. P(X=:c, Y:y)=ZP(X=:c, Y=y|N=n)P(N=n), where the summation is over all possible values of 71, holding a: and 3; xed. But unless n = 32+y, it is impossible for X to equal :6 and Y to equal 3;. For example, the only way there can be 5 hatched eggs and 6 unhatched eggs is when there are 11 eggs in total. So P(X=5, Y=6|N=n)=0 unless 'n, = 11, which means all other terms on the righthand side can be dropped, i.e., P(X=9:, Y=y) =P(X=:c, Y=y | N=m+y)P(N=x+y). Conditional on N = :1: + y, the events X = :1: and Y = y are exactly the same event, so keeping both is redundant; for example, if we know that {N = so + y} and {X = as}, then it is automatically the case that {Y = 3;}. Let us keep X = 3:. Then, P(X=a:, Yzy)=P(X=:c|N=3:+y)P(N=xly). Specify this joint probability mass function of X and Y, i.e., P(X = 3:, Y = y), with appropriate joint support of X and Y. Hint: We know that 0 S a: S 71, but an is removed (marginalized) during the derivation of the joint PDF of X and Y via the law of total probability. Then what are the possible values of :0 without the cap n? Similarly, what are the possible values of 3; without the cap n? (c) (2 pts) Based on the joint distribution of X and Y in (b), are X and Y independent? Explain. Hint: :2\" = eApeMlP). (d) (2 pts) Find Cov(N,X). Hint: N = X + Y as given in the problem, and so use the bilinearity of covariance. (e) (2 pts) Find Corr(N, X). Hint: Use the known facts about the marginal distribution of N and that of X