Consider the random variables x, y with joint distribution y \ x 0 1 2 0 1/9 2/9 1/9 1 2/9 0 2/9 2 0 1/9 0 1. (1 pts) find the marginal distributions of this joint distribution (i. e., compute the probability functions, p(x = x) and p(y = x), for all possible x). 2. (1 pts) find the conditional distribution of x given y = 1 (i. e., compute the probability function, p(x = x | y = 1), for all possible x). 3. (1 pts) are x and y independent? if yes, show that one of the equivalent characterizations of independence (given in class) is satisfied. if no, show that one of these characterizations is not satisfied.

a is the correct

if gloria has "carry over" minutes, (that is, any minutes paid for but not used in a month can be applied to her payment for minutes in the next month,) her total usage for the three months is: total minutes = 320 + 243 + 489 = 1052 minutes number of 200 minute blocks = 1052/200 = 5 r52 (that is 5 with a remainder of 52) rounding up the number of 200 minute blocks gives her usage for the three months as 6 blocks. the cost for six blocks is 6 x $25 = $150 she also has 200 - 52 or 148 carry over minutes if gloria pays each month and unused minutes are lost, applying the rounding up procedure to each month gives: first month usage of 320 minutes rounds up to 2 blocks of 200 minutes for a payment of 2 x $25 = $50 second month usage of 243 minutes rounds up to 2 blocks of 200 minutes for a payment of 2 x $25 = $50 third month usage of 489 minutes rounds up to 3 blocks of 200 minutes for a payment of 3 x $25 = $75 total payments for the three months are $50 + $50 + $75 = $175 get the plan with carry over minutes!