Wald's Equation - Statement

Statement

Let (Xn)n ∈ N be an infinite sequence of real-valued random variables and let N be a nonnegative integer-valued random variable. Assume that

1. (Xn)n ∈ N are all integrable (finite-mean) random variables,
2. E = E P(N ≥ n) for every natural number n, and
3. the infinite series satisfies

Then the random sums

are integrable and

If, in addition,

4. (Xn)n ∈ N all have the same expectation, and
5. N has finite expectation,

then

Remark: Usually, the name Wald's equation refers to this last equality.

Read more about this topic:  Wald's Equation

Famous quotes containing the word statement:

    “Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
    —Ralph Waldo Emerson (1803–1882)

    “The most distinct and beautiful statement of any truth must take at last the mathematical form.”
    —Henry David Thoreau (1817–1862)

    “The new statement is always hated by the old, and, to those dwelling in the old, comes like an abyss of skepticism.”
    —Ralph Waldo Emerson (1803–1882)