Random Variables Misc

Lazy Statistician

The rule of the lazy statistician states that if $Y=r(X)$, then

$$\mathbb{E}[Y]=\sum _{x\in S_X}r(x)f_X(x).$$

Continuous case:

$$\mathbb{E}[Y]=\mathbb{E}[r(X)]={\int }_{-\infty }^{\infty }r(x)f(x)dx.$$

Note the following points:

• The reason for the name “the rule of the lazy statistician” is that the rule is sometimes mistaken as a definition when, in fact, it is a statement that requires rigorous proof (we won’t do that here).
• The rule is sometimes called the law of the unconscious statistician (or LOTUS).
• In general, $\mathbb{E}[r(X)]\ne r(\mathbb{E}[X])$.