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Task 5 [Independent random variables} [} Let X and Y be independent random variables. Let f, g : IR 3- R be Bowl-measurable functions1 meaning that f'lf} E B[R}jg_l[C} E BUR} for any AC E BflR} Show that the random variables f [X :1 and gf} are independent. Task [} Let X be a random variable with density fx) = 1m: 1}21[_,,+1._,,+1] with 0:1 K E n1 :1 e n. {:1} Find K in terms ot'a. {b} Find the cumulative distribution function of X. {e} Calculate lP[% c: X E oz}. } [d Consider mix} := :3. 1'What is the probability density function of the random variable Y :2 SILK)