Function of is continuous and monotonically increasing

Question

Assignment:

1. Let x be a Poisson rv, with alphabet A_x = {0,1,2,...} and PMD P_x (a) = e^-Λ Λ^a/(a!).

a) Find the value of for which P [x = 2] is maximized.

b) Show that the probability P [x > 2] as a function of is continuous and monotonically increasing.

c) Calculate E [e_x].

2. Let x a Gaussian rv x~ N(0,σ²).

a) Find the PDF of y = x² + 1.

b) Calculate E x²(x + 1 + sin x).