Exercise 1 (Random variable and cdf) Let the probability space (0, F, P), where ? = {1, 2,3}, F = 2", and P satisfies P({1}) 0.5 and P({2}) = 0.1. Consider the random variable X : 0 - R defined by X (w) = 2w+3. (a) Find X-1([1, 5)) and X-1([2, 4] U {7}). (b) Derive the probability distribution of X as the induced measure PX-1. Be sure to define it for any Borel set B E B. (c) What is the support of X? (d) Derive the cumulative distribution function (cdf) of X. Be sure to define it for any rER. (e) Is X discrete or continuous? Justify