The random variable X has the following density function:

f(x) = { x^2 if 0 < x < 3
{ 0 otherwise

a) Find the distribution function F(x) of X
b) Draw the distribution function
c) Calculate the following probabilities:
P(X > 1.5) =
P(1/2 ≤ X < 3/2) =
P(-2 ≤ X < 1) =
d) What is E(X) and Var(X)? Do this calculation only as far as you can without a calculator

1 answer

well I guess if it looks like x^2 and the integral from 0 to 3 had to be all of it then it looks like
integral ax^2 dx from 0 to 3 = 1
a (3^3)/3 = 1
9 a = 1
a = (1/9)
so distribution is F(x) = x^2/9 defined only where x is between 0 and 3
now for example between 1.5 and 3 (your first domain)
integral (1/9) x^2 dx = (1/9)[ 3^3/3 - 1.5^3/3] = (1/27)[27-3.375 ] = .875
so probability that it is between 1.5 and 3 is 7/8