1. For each of the following, determine whether it can serve as the probability distribution of some random variable: a) p(k) = 1/7 , for k = 0; 1; 2; 3; 4; 5; b) p(x) = x2 /30 , for x = 0; 1; 2; 3; 4; c) p(y) = (y+4)/(y-4) , for y = 1; 2; 3; 4; 5. 2. Suppose that a random variable X can only take on values on the continuous interval from 0 to 4, and that its probability density function is given by: f(x) =x/8; for 0 \legslant x \legslant 4: a) Draw a graph of the probability density function. b) Proof that it is a proper density function. c) What is the probability that X will take on a value less than 2? d) What is the probability that X will take on a value greater than 3? e) What is the probablity that X will take on a value between 1 and 2.5? [Answer parts ( b) to (e) without doing any integration] 3. For Exercise 2 above, determine the mean and standard deviation. Locate the mean on your graph in Exercise 2.