step by step, detailed solution please

Question 1: The set of possible values of a discrete random variable X is {-3, 0, 3, 4}. p(x) the probability mass function of X is given by -3 0 3 4 p(2) 3/8 1/8 1/4 1/4 (a) Calculate E(X), Var(X), E(X2 - 3|X]), and E(X|X]). (b) Find the standardized X and its probability mass function. (25 points) Question 2: In a game, Emily gives Harry three well-balanced quarters to flip. Harry will get to keep all the ones that will land heads. He will return those landing tails. However, if all three coins land tails, Harry must pay Emily two dollars. Find the expected value and the variance of Harry's net gain. (15 points) Question 3: A box contains 10 disks of radii 1, 2, ..., 10, respectively. What is the expected value of the circumference of a disk selected at random from this box? (15 points) Question 4: Suppose that X is a discrete random variable with E(X) = 1 and E[X (X +2)] = 12. Find Var(-4X + 5) and o(-5X-2). (15 points) Question 5: Find the variance and the standard deviation of a random variable X with distribution function 0 x