Recall you did this problem in HW 15 Q5: Shankar is giving a one hour talk on hi

Question

Recall you did this problem in HW 15 Q5: 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

Px(0) = 2/10 , Px(1) = 3/10 , Px(2) = 3/10 , Px(3) = 2/10

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

Here we are given that P(X = 0) = 0.2, P(X = 1) = 0.3, P(X = 2) = 0.3 and P(X = 3) = 0.2

Also, the value of P(Y= 0) is computed as:

P(Y = 0) = P(X = 0) + P(Y = 0 | X = 1)P(X = 1) + P(Y = 0 | X = 2)P(X = 2) + P(Y = 0 | X = 3)P(X = 3)
P(Y = 0) = 0.2 + 0.3*0.2 + 0.3*0.22 + 0.2*0.23 = 0.2736

Now, we have:

P(X = Y = 0 ) = 0.2

Also, P(X = 0)P(Y = 0 ) = 0.2*0.2736 = 0.05472

Therefore, P(X = Y = 0 ) is not equal to P(X = 0)P(Y = 0 ) here.

Therefore X and Y are not independent here.

This was also intuitive by the fact that the number of questions correctly answered would depend on the number of questions asked.

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