1. (a)Suppose thatXBinomial(n, p). First write down the MGF (Moment Generating Function) ofX,MX(t). Prove that whenn andp0 butnp=, whereis a positive constant, we haveMX(t) converges to the MGF ofP oisson().
2. (b)Part (a) essentially states that a Binomial random variable X, such thatXBinomial(n,p), can be approximated by a Poisson random variable Y with parameterwhennis large,pis small and=np. Now, let's investigate this statement.

• i. First suppose n is small, with n=10, p=0.3and =3. CalculateP(X=3) under theBinomial(10,0.3) distribution, and also calculateP(Y= 3) under thePoisson(3) distribution. Calculate the absolute difference of these two probabilities, that is|P(X= 3)P(Y= 3)|.
• ii. Next, we consider a slightly largern, withn= 100,p= 0.03 and again= 3. CalculateP(X= 3) under theBinomial(100,0.03) distribution, and also calculateP(Y= 3) under theP oisson(3) distribution. Calculate the absolute difference of these two probabilities, that is|P(X= 3)P(Y= 3)|.
• iii. Lastly, we consider a largen, withn= 1000,p= 0.003 and again= 3. CalculateP(X= 3) under theBinomial(1000,0.003) distribution, and also calculateP(Y= 3) under theP oisson(3) distribution. Calculate the absolute difference of these two probabilities, that is

|P(X= 3)P(Y= 3)|.

• iv. What did you find based on the absolute differences you calculated in (i), (ii), and (iii)? Do these numbers agree with the result in part (a)?

(a) (6 points) Suppose that X ~ Binomial (n, 13). First write down the MGF (Moment Generating Function) of X, M X (t). Prove that when n > 00 and p > 0 but up = A, where A is a positive constant, we have M X (t) converges to the MGF of Poisson(A). (Hint: 11mm\