Question: Exercise 1 (Random variable and cdf) Let the probability space (0, F, P), where 0 = {1, 2, 3}, F = 2, and P satisfies

Exercise 1 (Random variable and cdf) Let the probability space (0, F, P), where 0 = {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, 5)) and X-'([2, 4] U {7}). (b) Derive the probability distribution of X as the induced measure PX-. 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 TER. (e) Is X discrete or continuous? Justify

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