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Establish the validity of the expected value rule E [g(X)]
Chapter , Problem 4(choose chapter or problem)
Establish the validity of the expected value rule E [g(X)] = I: g(x)fx (x) dx, where X is a continuous random variable with PDF fx . 185 Solution. Let us express the function 9 as the difference of two nonnegative functions, where g+ (x) = max{g(x) , O}, and g- (x) = max{ -g(x), O}. In particular. for any t 2::: 0, we have g(x) > t if and only if g+(x) > t. We will use the result E [g(X)] = f.X> p(g(X) > t) dt - f.X> p(g(X) < -t) dt from the preceding problem. The first term in the right-hand side is equal to 100 f fx (X) dX dt = j oo f fx (X) dt dx = j = g+ (x)fx (x) dx. o i{x I g(x t} -00 i{t l 0:$ t By a symmetrical argument, the second term in the right-hand side is given by f.= p(g(X) < -t) dt = I: g-(x)fx (x) dx. Combining the above equalities, we obtain E [g(X)] = I: g+ (x)fx (x) dx - I: g- (x)fx (x) dx = I: g(x)fx (x) dx.
Questions & Answers
QUESTION:
Establish the validity of the expected value rule E [g(X)] = I: g(x)fx (x) dx, where X is a continuous random variable with PDF fx . 185 Solution. Let us express the function 9 as the difference of two nonnegative functions, where g+ (x) = max{g(x) , O}, and g- (x) = max{ -g(x), O}. In particular. for any t 2::: 0, we have g(x) > t if and only if g+(x) > t. We will use the result E [g(X)] = f.X> p(g(X) > t) dt - f.X> p(g(X) < -t) dt from the preceding problem. The first term in the right-hand side is equal to 100 f fx (X) dX dt = j oo f fx (X) dt dx = j = g+ (x)fx (x) dx. o i{x I g(x t} -00 i{t l 0:$ t By a symmetrical argument, the second term in the right-hand side is given by f.= p(g(X) < -t) dt = I: g-(x)fx (x) dx. Combining the above equalities, we obtain E [g(X)] = I: g+ (x)fx (x) dx - I: g- (x)fx (x) dx = I: g(x)fx (x) dx.
ANSWER:Step 1 of 2
Assume that function g(x) is expressed using two other positive functions as follows,
Such that,
