2. Suppose Y is a continuous random variable with probability density function given by fY (y) = 1/2 |y|/4, 2 y 20, elsewhere.

(a) Sketch the probability density function of Y .

(b) From your sketch, or otherwise, verify that this is indeed a valid pdf.

(c) Evaluate the following probabilities:i. P(0 Y 0.4);ii. P(Y > 0.5);iii. P(Y = 0);iv. P(Y 0.5).

(d) Suppose that X1 and X2 are independent Uniform[1, 1] random variables. It can be shown that X1 +X2 has the same probability density function as Y ; i.e.that X1 + X2 and Y are identically distributed. Using this fact, and the facts that the mean and variance of the Uniform[a, b] distribution are (a + b)/2 and(b a)2/12 respectively, find the following:

i. P(X1 + X2 0.5);

ii. E[Y ] and var(Y ).