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.

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