me duality. 6. (BH 6.19) Use MGFs to determine whether X + 2Y is Poisson if X and Y are i.i.d. Pois()). 7. (BH 6.22) Consider a setting where a Poisson approximation should work well: let Al, ..., An be independent, rare events, with n large and p; = P(A;) small for all j. Let X = I(Al) + . . . + I(An) count how many of the rare events occur, and let 1 = E(X). (a) Find the MGF of X. (b) If the approximation 1 + x ~er (this is a good approximation when a is very close to 0 but terrible when x is not close to 0) is used to write each factor in the MGF of X as e to a power, what happens to the MGF? Explain why the result makes sense intuitively. ro.html ype here to search 480 F ~ ( 1) ENG US ALIENWARE