A random variable X has the cumulative distribution function
Calculate the variance of X.
The answer specifies that the density function is
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.
Another way of thinking about this is to think of the density as a generalised function, then we may represent the density of a piecewise continuous cumulative distribution as the sum of a continuous function and delta functionals at the discontiuities of amplitude equal to the size of the jumps.