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.
I’ll use today’s one-liner whena step that’s usually necessary in a calculation isn’t needed for a particular example. For example, consider the following problem from probability:
Let be uniformly distributed on . Find .
The first step is to write . Then we start computing the expectations. To begin,
Ordinarily, the next step would be computing . However, this computation is unnecessary since will be multiplied by , which we just showed was equal to . While I might calculate if I thought my class needed the extra practice with computing expectations, the answer will not ultimately affect the final answer. Hence my one-liner:
To paraphrase the great philosopher The Rock, it doesn’t matter what is.
P.S. This example illustrates that the covariance of two dependent random variables ( and ) can be zero. If two random variables are independent, then the covariance must be zero. But the reverse implication is false.