In n+m independent Bernoulli(p) trials, let S_n be the number of successes in the first n

trials, T_m the number of successes in the last m trials.

(A) What is the distribution of S_n? Why?

Requirement for solution for why part:

You are asked to prove it for the general case. A proof for a special case is given in Lecture note 4. If you cannot give a proof for the general case, you should give a proof for a special case, say, n=5, r=3, p=0.2.

(B) What is the distribution of T_m? Why?

Requirement: Write down your formula. One or two sentences to explain why.

(C) What is the distribution of S_n+T_m ? Why?

Requirement: Write down your formula. One or two sentences to explain why

(D) Are S_n and T_m independent? Why?

Requirement: Give your answer and one or two sentences to explain why. You don't need to prove it mathematically. Use of the common sense should be ok.

Please answer the above questions and I will give you an upvote, thankyou:)!