Verify the expected value rule E[g(X, Y)] = L

Chapter , Problem 35

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QUESTION:

Verify the expected value rule E[g(X, Y)] = L Lg(x,y)pX,y(x,y), x y using the expected value rule for a function of a single random variable. Then, use the rule for the special case of a linear function, to verify the formula E[aX + bY] = aE[X] + bErYl , where a and b are given scalars. Solution. We use the total expectation theorem to reduce the problem to the case of a single random variable. In particular, we have E[g(X, Y)] = LPy(Y)E[g(X, Y) I Y = Y] y = LPy(y)E[g(X, y) I Y = Y] y y x = L L g(x, y)px.y(x, y), x y as desired. Note that the third equality above used the expected value rule for the function g(X, y) of a single random variable X. For the linear special case, the expected value rule gives E[aX + bY] = L L (ax + by )px. y (x, y) x y = a LX Lpx.y(x,y) + b LY Lpx.y(x,y) x y y x = a LXPx(x) + b LYPY(Y) x y = aE[X] + bErYl .

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QUESTION:

Verify the expected value rule E[g(X, Y)] = L Lg(x,y)pX,y(x,y), x y using the expected value rule for a function of a single random variable. Then, use the rule for the special case of a linear function, to verify the formula E[aX + bY] = aE[X] + bErYl , where a and b are given scalars. Solution. We use the total expectation theorem to reduce the problem to the case of a single random variable. In particular, we have E[g(X, Y)] = LPy(Y)E[g(X, Y) I Y = Y] y = LPy(y)E[g(X, y) I Y = Y] y y x = L L g(x, y)px.y(x, y), x y as desired. Note that the third equality above used the expected value rule for the function g(X, y) of a single random variable X. For the linear special case, the expected value rule gives E[aX + bY] = L L (ax + by )px. y (x, y) x y = a LX Lpx.y(x,y) + b LY Lpx.y(x,y) x y y x = a LXPx(x) + b LYPY(Y) x y = aE[X] + bErYl .

ANSWER:

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The expected value rule for a function of a single random variable is determined by,

                                      

Since

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Then,

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