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6. Two dice are rolled.

A = 'sum of two dice equals 3'

B = 'sum of two dice equals 7'

C = 'at least one of the dice shows a 1'

(a) What is P(A|C)? (b) What is P(B|C)? (c) Are A and C independent? What about B and C?

7. A multiple choice exam has 4 choices for each question. A student has studied enough so that the probability they will know the answer to a question is 0.5, the probability that they will be able to eliminate one choice is 0.25, otherwise all 4 choices seem equally plausible. If they know the answer they will get the question right. If not they have to guess from the 3 or 4 choices. As the teacher you want the test to measure what the student knows. If the student answers a question correctly what's the probability they knew the answer

8.Suppose that buses arrive are scheduled to arrive at a bus stop at noon but are always X minutes late, where X is an exponential random variable with probability density function fX(x) = ex. Suppose that you arrive at the bus stop precisely at noon. (a) Compute the probability that you have to wait for more than five minutes for the bus to arrive. (b) Suppose that you have already waiting for 10 minutes. Compute the probability that you have to wait an additional five minutes or more.

9. . Transforming Normal Distributions Z Suppose Z N(0,1) and Y = e . (a) Find the cdf FY (a) and pdf fY (y) for Y . (For the CDF, the best you can do is write it in terms of the standard normal cdf.) (b) We don't have a formula for (z) so we don't have a formula for quantiles. So we have to write quantiles in terms of 1. (i) Write the 0.33 quantile of Z in terms of 1 (ii) Write the 0.9 quatntile of Y in terms of 1. (iii) Find the median of Y .

10. Let X and Y be two continuous random variables with joint pdf 2 f(x, y) = cx y(1 + y) for 0 x 3 and 0 y 3, and f(x, y) = 0 otherwise. (a) Find the value of c. (b) Find the probability P(1 X 2, 0 Y 1). (c) Determine the joint cdf of X and Y for a and b between 0 and 3. (d) Find marginal cdf FX(a) for a between 0 and 1. (e) Find the marginal pdf fX(x) directly from f(x, y) and check that it is the derivative of FX(x).