1. Let X and Y be continuous random variables with joint distribution f(x, y) = cxy on the triangle formed by the points (0, 0),(1, 0), and (1, 1).
(a) Find c.
(b) What is the probability that x < 1 2 ?
(c) Find the marginal distribution of X, fX(x)?
2. Let X and Y have the same variance. What is Cov(X + Y, XY )?
3. Suppose Alice takes the bus to work when the weather is clear, and drives her car to work when it is raining. The probability of rain is 20%. The wait time for the bus follows an exponential distribution with an average wait time of 5 minutes. Once the bus arrives it always takes 35 minutes to get to work. When Alice drives to work, her commute time is normally distributed with a mean of 30 and standard deviation of
5. (a) What is the theoretical mean of Alices commute time?
(b) Use simulations to estimate the mean and variance of Alices commute time.
(c) Estimate by simulation the probability that the bus trip, including
the wait, is longer than the car trip.