Let {X1, X2,...} be a sequence of independent Poisson random variables, each with parameter 1. By applying the central limit theorem to this sequence, prove that lim n 1 en .n k=0 nk k! = 1 2 .

ST 701 Week Four Notes MaLyn Lawhorn September 5, 2017 and September 7, 2017 Independence In words, events A and B are said to be independent if knowledge that B occurred doesn’t change our uncertainty about A. Mathematically, this can be written as if P(B) > 0, then A and B are independent events if P(AjB) = P(A). However, this deﬁnition makes it appear as if direction matters. It sounds as if A being independent from B is not the same as B being independent from A, but this is not the case. A more formal, but less intuitive, mathematical deﬁnition of independence is if A and B are in- dependent events, P(A [ B) = P(A)P(B). In many problems, we can associate the word inde- pendence with the action of multiplying probabilitie