PROBLEM 1 (10 points). Let X be a random variable sampled uniformly from [0,1] (i.e., X is equal to a number sampled uniformly at random from that interval). (a) Is X a discrete or continuous random variable? Justify your answer. (b) Find the probability that X < 0.3. (c) Find the probability that X is in the set {0.1, 0.2, 0.3, 0.4}. (d) Find the probability that X is irrational. Let X be the first decimal digit of X. (e) Find the range of X, i.e., its set of possible values. Is X a discrete or a continuous random variable? (f) Find the PDF and CDF of X1. Let Y be a second random variable that is sampled uniformly from the interval [0,1] (X and Y are sampled independently). (g) Find the probability that X > Y. Hint: Think of the 2D region of points (X, Y). Solution: Your solution here. PROBLEM 2 (10 points). We have a 4-sided die with the numbers 1,2,3,4 written on its faces. We roll the die twice. Consider the following random variables: Let M be the maximum of the values of the two rolls. Let R be the value of the first roll. (a) Find the PDF and CDF of M. (b) Find the probability that M is odd. (c) Find the probability that M is odd or R is even. (d) Find the probability that M = R.