Question

Let X be given by its distribution function F(x) , such that
f(x) = 0 if x<=0
f(x) = 1/16 x^4, if 0 < x <=2
f(x) = 1 , if x > 2
1. Graph the distribution function F(x)
2. Graph the density function
3. Find E(X), Var(X), SD(X)

Answer 1

To find the density function, differentiate each part of the density function.\[E(X)=\int\limits_{-\infty}^{\infty}xf(x)dx\]Because f(x) is only greater than zero when 0<x<2, you can just use these as your limits of integration instead of infinity.\[Var(X)=E(X^2)-[E(X)]^2=\int\limits_{-\infty}^{\infty}x^2f(x)dx-[E(X)]^2\]Again because your function is only greater than zero when 0<x<2, you can just use zero as your lower and two as your upper limits of integration.\[SD(X)=\sqrt{Var(X)}\]

Answer 2

Sorry, the first line should be: To find the density function, differentiate each part of the DISTRIBUTION function.