1. Consider a coin that has a probability of getting heads when it is tossed. Assume that you tossed the coin repeatedly until you observe the pattern T, T, H. More specifically, you stop tossing the coin when you observe a sequence of two tails followed by a head. Let X denote the number of tosses made. Find the expected value of X.

2. First a point is selected at random from the interval (0,1). Then another point is selected at random from the interval in such a way so that . Find the probability density function of .