1. You arrive at a bus stop at 10 o'clock, knowing that the bus will arrivve at some time uniformly distributed between 10 and 10:30.
    1. What is the probability that you will have to wait longer than 10 minutes?
    2. If at 10:15 the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

  2. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter 1/2. What is
    1. the probability that the repair time exceeds 2 hours;
    2. the conditional probability that a repair takes at least 10 hours, given that its duraction exceeds 9 hours?

  3. The joint probability density function of X and Y is given by

    f(x,y)=(6/7)(x2 + (xy)/2)     0<x<1, 0<y<2

    1. Find P(X>Y).
    2. Find P(Y>1/2 | X<1/2).

  4. A guy and a girl agree to meet at Perelman Quad about 12:30 PM. If the guy is equally likely to arrive anytime between 12:15 and 12:45 and the girl independently arrives at a time uniformly distributed between 12 and 1, find
    1. the probability that the first to arrive waits no longer than 5 minutes;
    2. the probability that the guy arrives first.