Consider a random variable X whose distribution function (cdf) is given by FX(x)

Question

Consider a random variable X whose distribution function (cdf) is given by
FX(x) =
8>>>>><>>>>>:
0 if x < ?2
0.2 if ?2 x < 2.2
0.25 if 2.2 x < 4
0.5 if 4 x < 5
0.6 if 5 x < 2
1 if x 2
(a). (4 points) Give the probability mass function, p(x), of X, explicitly.
(b). (3 points) Compute P(?2 < X < 3).
(c). (3 points) Compute P(X 3 and X is rational).
(d). (3 points) Compute P(X 4 | X 0).
(e). (6 points) What is the cdf (cumulative distribution function) of Y = X2? (be explicit!)
(f). (6 points) Compute E(|X|). Also compute E(p(X)F(?X)),

Explanation / Answer

For the first problem you would integrate f(x) on the interval (0,2) and set it equal to 1 since the area under the curve is 1. c(x^2 - x^3 /3) evaluated from 0 to 2 c(4 - 8/3) c(12 - 8)/3 c(4/3) so c = 3/4 You already have the cdf function from what you've already computed. The integral of a pdf is the cdf. F(x) = (3/4)(x^2 - x^3 /3) To fill in the missing blanks it's supposed to say 0 if x

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