Let X be a continuous random variable that takes values in [0, 1], a.pdf

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Let X be a continuous random variable that takes values in [0, 1], and whose cumulative distribution function F satis?es F(x) = 2x^2 - x^4 for 0 <= x <= 1. (a) Compute P(1/4 <= X <= 3/4). (b) What is the probability density function of X? Solution F(x) = 2x^2 - x^4 for 0 x 1 a) P(1/4 X 3/4) = F(3/4) - F(1/4) F(3/4) = 2(.75)^2 - (.75)^4 = 1.125 - 3.1640625 = .80859375 F(1/4) = 2(.25)^2 - (.25)^4 = .125 - .00331776 = .12168224 F(3/4) - F(1/4) = .80859375 - .12168224 = .68691151 or approximately .6869 b) pdf = f(x) = the derivative of 2x^2 - x^4 = 4x - 4x^3 from 0 x 1..

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