Choose a number x at random from the set of numbers {1, 2, 3, 4, 5}. now choose a number at random from the subset no larger than x, that is, from {1, . . ,x}. call this second number y. (a) find the joint mass function of x and y. (b) find the conditional mass function of x given that y = i. do it for i = 1, 2, 3, 4, 5. (c) are x and y independent? why?

a) for any k in {1,2,3,4,5}

for any j n {1,2,3,4,5}

c) No

Step-by-step explanation:

For X, we have 1 chance out of five to get any of the five numbers, so for any integer k between 1 and 5.

For Y, we can use the theorem of total probability, because we can divide Y in five different cases depending on how it went with the value of X. We can remove impossible cases (of probability 0), those cases happen when X is smaller than the value we want to get with Y.

b) In order to solve this item, we should use Bayes Formula. Note that for any i, j we have

This probability is always 0 if j is smaller than i. If j is bigger or equal than i, then P(Y = i |X=j) is 1/j. Also P(X=j) = 0.2. Therefore, for i ≤ j, we have

Since we calculated the values of P(Y=i), we alredy know this values

c) X and Y arent independent because if they were, we should have that P(X=j) shoudnt change if we condition anything from Y, but P(X=j) = 0.2, and we obtained results different from 0.2 when we conditioned by different values of Y in the previous item.