In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.
Here’s a problem that could appear in my class in probability or statistics:
Let be a probability density function for . Find , the cumulative distribution function of .
A student’s first reaction might be to set up the integral as
The problem with this set-up, of course, is that the letter has already been reserved as the right endpoint for this definite integral. Therefore, inside the integral, we should choose any other letter — just not — as the dummy variable.
Which sets up my one-liner: “In the words of the great philosopher Jean-Luc Picard: Plenty of letters left in the alphabet.”
We then write the integral as something like
and then get on with the business of finding .