Lovely Indicator Function Behaviour

published on 2026-10-10

I think I personally just slept through Intro to Mathematical Statistics, but what lovely behaviour.

For a random variable $X$: $$ E[\mathbf 1(X = x)] = P(X=x) $$

Since if we just expand: $$ E[\mathbf 1(X = x)] = 1 \times P(X=x) + 0 \times P(X\neq x) = P(X=x) $$

And, you could define a CDF like: $$ F(x) = E[\mathbf 1(X\leq x)] $$

How nice!