Solve all three. I'll upvote

1. Derive the characteristics function of X given that ifr 2 2. Use the characteristics function of X to derive the variance of X given that 1/3 ifx=0 1/3 ifx=1 Px(X) = 1/3 if x = 2 0 otherwise 3. If An = {n, n + 1, .. .} with P({n}) = 4 n(n+1); n = 1, 2. .... . Find: a) En P(An). b) P(lim, -+ x An). . Can we apply the Borel-Cantelli Lemma to find P(lim An)? Explain your an- swer