Consider the following two random variables: ˜ X has a (continuous) uni form density on [−1,+1], while ˜ Y is a discrete random variable defined by (−1, 1 2;+1, 1 2).

(a) Do ˜ X and ˜ Y have the same mean?

(b) Compute their variances.

(c) Draw their cumulative distributions.

(d) Which random variable is riskier? Apply the ‘integral condition’ and also ask yourself which random variable has more weight in the center.

(e) Find the distributions of the ‘white noise’that must be added to the less risky lottery to obtain the riskier one.