Discrete version

Suppose is a discrete r.v. and . By definition, the Expectation . However, this requires us to find , the PMF of . Instead, if we already have the PMF of , we can use that instead:

Continuous version

This formula is even more useful in the continuous version, because finding the PDF of given ‘s PDF is often very nontrivial.

Often, when we’re told to not use the PDF when finding expectation, this just means to use LOTUS instead.

Suppose is a continuous r.v. and , where . Again, . We want to avoid finding , so instead we can calculate as follows, given we know .

Calculating variance

This formula is useful when we want to calculate the Variance of and need . We simply let and so

2D version

Consider two r.v.s and with different distributions, and let their Joint distribution be . Another way of interpreting this is as a transformation . If we wanted to find , calculating its PDF may be difficult. Instead, we claim that

for two discrete distributions. If and are both continuous and their joint PDF is , then we have