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Computing expectation by conditioning. Let (S, E, P) be a probability space, let F E be such that
 

Summary: Computing expectation by conditioning.
Let (S, E, P) be a probability space, let F E be such that
P(F) > 0.
Suppose X is a discrete random variable. We let
E(X|F)
be the expectation of X with respect to the probability P(|F). It follows that
(1) E(X|F) =
x
xP(X = x|F).
Moreover, if F1, . . . , Fn E are such that Fi Fj = whenever i = j and S = n
i=1Fi then, as one may
easily verify,
(2) E(X) =
n
i=1
E(X|Fi)P(Fi).
Example. Let X1, X2, X3, . . . be a sequence of independent Bernoulli random variable with parameter
p > 0. Let q = 1 - 0 and let
G = min{n : Xn = 0}
so G is geometric with parameter p.

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics