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.

• Firstsupposenissmall,withn=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)|.
• 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)|.
• 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)|.
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(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\