A cubic distribution function

Topic: Random variables and distributions · more questions on this topic
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A continuous random variable X has distribution function F(x)=0 for x<0, F(x)=x^3 for x\in[0,1], and F(x)=1 for x>1.

  1. Find the density of X.

  2. Calculate \mathbb{P}(1/2<X<3/4).

  3. Calculate \mathbb{E} X.

  1. f(x)=F'(x)=3x^2 for 0<x<1, and f(x)=0 elsewhere.

  2. \mathbb{P}(1/2<X<3/4)=F(3/4)-F(1/2)=(3/4)^3-(1/2)^3=19/64.

  3. \mathbb{E} X=\int_0^1 x\cdot 3x^2\,\,\mathrm{d} x=3/4.