Alternate title: Claudia Makes Things Way More Complicated than They Need to Be Because She Sucks

We had this bonus question on our homework for Probability today:

Suppose X has a density defined by

Let F_X(x) be the cumulative distribution of X. Find the area of the region bounded by the x-axis, the y-axis, the line y = 1, and the curve y = F_X(x).

And I was like, “Aw, sweet! Areas of regions! CALCULUS!”

So first, I had to find the cumulative distribution function (cdf) of X. Easy. It’s just the integral of the density f_X(x) from negative infinity to a constant b. In this case:

With 2 ≤ b ≤ 3. So that’s my curve y. The area I’m looking for, therefore, is this (the red part, not the purplish part):

Now anyone with half a brain would look at this and go, “oh yeah, that’s easy. I can find the area of the rectangle formed by the two axes, the line y = 1, and the line x = 3, then find the area of the region below the curve from 2 to 3, and subtract the latter from the former to get the correct area.”

Which works. Area of rectangle = 3, area of region below F_X(x) = .25, area of region of
interest = 2.75.

Or they could remember the freaking formula that was explicitly taught last week. Such areas can be calculated using:

But did I see either of those? Nooooooope.

I looked at the graph and was like, “how the hell do you find that?” I tried a few things that didn’t work, then realized that it would be a lot easier to figure out if I changed the integral from being in terms of x (or b, rather) to being in terms of y.

So then I just had to integrate. This gave me the right answer: 2.75!

Moral of the story: don’t complicate things. But if you do complicate things, you might actually end up in a scenario where you’ll use something that you were taught back in calculus I but didn’t ever suspect you’d actually use. I had appreciated learning the handy-dandy technique of changing variables, but I didn’t think I’d be in a situation where I’d apply it. Shows what I know, eh?

It was a nice refresher, at least. I’ve missed calculus.