A random variable X has the cumulative distribution function
for
for
for
Calculate the variance of X.
The answer specifies that the density fucntion is
if x=1
if 1<x<2
otherwise
Then
I got the f(x) = x-1 part, and I got how to calculate the variance after you have the expected values, but I'm lost on other questions.
My questions are:
Where do we get if x=1? F(1) = 0.5, but I can't figure out why f(1) would equal 0.5.
What is the rule for putting parts of the stepwise density function into the expected value equations? I don't know what the rule is called so I don't know how to review it. We're adding the slope at a single point to the slope over a big area, which is something I can't quite work out visually.