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More on expectation and random variables. Definition. Suppose (S, E, P) is a probability space and X : S R is a random variable. Suppose
 

Summary: More on expectation and random variables.
Definition. Suppose (S, E, P) is a probability space and X : S R is a random variable. Suppose
g : R R.
Let
g(X) : R R
be such that
g(X)(s) = g(X(s)) whenever s S.
Thus g(X) is g X where is composition of functions. If g is a Borel function then g(X) is a random
variable. Any g's we deal with will be Borel function. I omit the technical definition of Borel function.
Theorem. Suppose (S, E, P) is a probability space, X : S R is a random variable and
g : R R
is a Borel function. Then
E(g(X)) =
xR
g(x)pX (x).
Proof. We prove this in the case where the range of X is finite, say rng X = {x1, . . . , xm} where x1 <
< xm. Then
E(g(X)) =
sS
g(X(s))P({s})

  

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

 

Collections: Mathematics