Tail-Sum Formula

For non-negative variables, the mean is the sum (or integral) of tail probabilities.

What is it?

For a non-negative integer variable, $E [X] = \sum_{k \geq 1} P (X \geq k)$ ; for continuous, $E [X] = \int_{0}^{\infty} P (X > x) d x$ .

Tails of minima and waiting times are often simple products, while the pmf is messy.

Use it when $P (X \geq k)$ is easier than $P (X = k)$ , typically for minima and waiting times.

Think of this method when…

Each value $k$ is counted once for every threshold it exceeds, so summing tails recovers the mean.

E [X] = k = 1 \sum \infty P (X \geq k)

Two fair dice are rolled. What is the expected minimum?

1
$P (min \geq k) = (\frac{7 - k}{6})^{2}$ for $k = 1, \dots, 6$ .
2
Sum the tails.
$\frac{36 + 25 + 16 + 9 + 4 + 1}{36} = \frac{91}{36} \approx 2.53$
Answer: $91/36$

Is the tail-sum formula a good first step for the variance of a normal variable?

8 QuantSolve questions are tagged with this method.