Wald's equation

In probability theory, Wald's equation is an important identity which simplifies the calculation of the expected value of the sum of a random number of random quantities. Formally, it relates the expectation of a sum of randomly many i.i.d. random variables to the expected number of terms in the sum and the random variables' common expectation.

Let "X"₁, "X"₂, ..., "X"_"T" be a sequence of "T" i.i.d. random variables distributed identically to some random variable "X", such that
# "T" > 0 is itself a random variable (integer-valued),
# the expectation of "X", E("X") < ∞, and
# E("T") < ∞.

Then::$$operatorname{E}left(sum_{i=1}^{T}X_i ight)=operatorname{E}(T)operatorname{E}(X).

In this case, the random number "T" acts as a stopping time for the stochastic process { "X"_"i", "i" = 1, 2, ... }.

Proof

Define a second sequence of random variables, "Y_n":

$$Y_n = sum_{i=1}^{n}X_i - noperatorname{E}(X)

It can be seen from elementary probability that "Y_n" is a martingale, and moreover satisfies the conditions of the optional stopping theorem. Hence

$$operatorname{E}left(sum_{i=1}^{T}X_i - Toperatorname{E}(X) ight) = operatorname{E}(Y_T) = operatorname{E}(Y_0) = 0

And the result follows by simple rearrangement.

Proof #2

Let X_i be independent and identically distributed (iid) variables with mean μ_X , and N a non-negative random number with mean μ_N independent of all X_i. Define Y:

$$Y = sum_{i=1}^{N}X_i

Then the expectation of Y, E [Y] is:

$$E [Y] = E left [ sum_{i=1}^{N}X_i ight]

Condition the expectation on the variable N:

$$egin{align} E [Y] &= sum_{n=0}^infty E left [ sum_{i=1}^{n} X_i ~ | ~ N=n ight] P_N(n) \ &= sum_{n=0}^infty left( sum_{i=1}^{n}E [X_i ~ | ~ N=n] ight) P_N(n) \ &= sum_{n=0}^infty left( sum_{i=1}^{n}E [X_i] ight) P_N(n) \ &= sum_{n=0}^infty n mu_X P_N(n) \ &= mu_X sum_{n=0}^infty n P_N(n) \ &= mu_X mu_N end{align}

The the second line is due to the linearity of the conditional expectation, the third line due to the independence of X_i and N.

ee also

*Abraham Wald

References

*cite journal|last = Wald|first = Abraham|title = On Cumulative Sums of Random Variables|journal = The Annals of Mathematical Statistics|volume = 15|issue = 3|date = Sep 1944|pages = 283–296|url = http://links.jstor.org/sici?sici=0003-4851%28194409%2915%3A3%3C283%3AOCSORV%3E2.0.CO%3B2-0&size=LARGE|doi = 10.1214/aoms/1177731235|month = Sep|year = 1944
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