Recall you did this problem in HW 18: Shankar is giving a one hour talk on his r

Question

Recall you did this problem in HW 18: Shankar is giving a one hour talk on his research at a big university. If X denotes the number of questions he is asked by audience members in his talk then suppose X has probability mass function p_X (0) = 1/10, P_X (1) = 4/10, P_X (2) = 4/10, P_X (3) = 1/10 Every question he gets, Shankar is able to adequately answer with probability 4/5, independent across questions. Let Y denote the total number of "adequately" answered questions. Are X and Y independent? Give a math justification. Just putting down the answer will not get you full credit.

Explanation / Answer

Back-up Theory

If A and B are independent, P(A B) = P(A) x P(B)

Now, to work out solution,

We need to find the joint probability of X and Y and the the pmf of X and Y. pmf of X is already given. To find pmf of Y, Y can take values 0, 1, 2 and 3 and probability of ‘adequately’ answering a question is 4/5 and is independent across questions. So, Y ~ B(3, 4/5).

Now, P(X = 1, Y = 1)

= probability that only one question is put and Shankar answers it adequately = (4/10)x(4/5) = 16/50.

P(X = 1) = 4/10 and P(Y = 1) = 3C1(4/5)1(1/5)2 = 12/125.

P(X = 1) x P(Y = 1) = 48/1250 16/50 => X and Y are not independent. ANSWER

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