Hope everyone is well. I'm wondering if someone can help me with a difficult problem presented during lecture today.
Two friends plan to meet to go to a nightclub. Each of them arrives at a time uniformly distributed between midnight and 1am and independently of the other. Denote by X (respectively Y) the random variable representing the arrival time of the first person (respectively, the second). The joint probability distribution is given by
f_(x,y) (x,y) = 2 if 0 ≤ x ≤ y, and 0 otherwise.
a) Find the probability that the first person is waiting for his friend for more than 10 minutes.
b) Determine the marginal probability density functions of X and Y. Check that they are indeed probability density functions.
c) Calculate the means E[X] and E[Y].
d) Calculate the variances V(X) and V(Y).
e) Find the conditional density function of X given that Y = y, for 0 ≤ y ≤ 1. Check that it is indeed a probability density function.
f) Repeat part (e) for the conditional density of Y given that X = x.