Tail-Sum Formula

For a nonnegative integer-valued X, E[X] = Σ P(X ≥ k) — often far easier than summing k·P(X=k) directly.

1. The idea

For a random variable $X$ taking values in ${0, 1, 2, \dots}$ , the expectation can be rewritten as $E [X] = \sum_{k = 1}^{\infty} P (X \geq k)$ . This avoids computing individual pmf values and instead sums survival probabilities.

The formula is especially powerful for waiting times, minima, and maxima, where $P (X \geq k)$ has a clean closed form (e.g., 'no success in the first $k - 1$ trials') even though $P (X = k)$ is messier to combine directly.

It follows from swapping the order of summation in $E [X] = \sum_{k} k P (X = k)$ , writing $k$ as a sum of 1's counted once for each $j \leq k$ .

2. Why it matters

Tail sums turn waiting-time and extreme-value expectation problems — very common in interviews — into a single geometric-style series instead of an awkward direct sum.

3. Core mathematics

Tail-sum formula

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

X nonnegative, integer-valued.

Continuous analogue

E [X] = \int_{0}^{\infty} P (X > x) d x

X nonnegative, continuous.

4. How to recognize it

Think of this when…

X counts trials until an event, or is a minimum/maximum of several variables
P(X ≥ k) or P(X > x) has a clean form, but the pmf is awkward
The question involves waiting times or 'first time that...' language

5. Worked example

Worked example

Let $X$ be geometric with success probability $p$ on ${1, 2, 3, \dots}$ (number of trials until first success). Use the tail-sum formula to find $E [X]$ .

1
$P (X \geq k)$ is the probability of at least $k - 1$ failures first.
$P (X \geq k) = (1 - p)^{k - 1}$
2
Sum the geometric series over $k = 1, 2, 3, \dots$ .
$E [X] = k = 1 \sum \infty (1 - p)^{k - 1} = \frac{1}{1 - ( 1 - p )}$
3
Simplify.
$E [X] = \frac{1}{p}$
Answer: $E [X] = 1/ p$ , matching the known geometric mean

6. Common mistakes

Starting the sum at $k = 0$ instead of $k = 1$ (or mishandling the boundary convention)
Confusing $P (X \geq k)$ with $P (X = k)$ when applying the formula
Applying the discrete formula to a continuous variable without switching to the integral form
Forgetting the formula requires $X \geq 0$

7. Quick check

For $X$ uniform on ${1, 2, 3}$ , using the tail-sum formula, what is $E [X]$ ?

8. Practice

28 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Tail-Sum Formula

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