Their joint pdf is the product of their individual

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Unformatted text preview: ts are placed on the function g. Suppose you observe X = 10. What do you know about Y ? Well, if you know the joint pdf of X and Y , you also know or can derive the conditional pdf of Y given X = 10, denoted by fY |X (v |10). Based on the fact, discussed above, that the minimum MSE constant estimator for a random variable is its mean, it makes sense to estimate Y by the conditional mean: ∞ E [Y |X = 10] = vfY |X (v |10)dv. −∞ The resulting conditional MSE is the variance of Y , computed using the conditional distribution of Y given X = 10. E [(Y − E [Y |X = 10])2 |X = 10] = E [Y 2 |X = 10] − (E [Y |X = 10]2 ). 4.10. MINIMUM MEAN SQUARE ERROR ESTIMATION 163 Conditional expectation indeed gives the optimal estimator, as we show now. Recall that fX,Y (u, v ) = fX (u)fY |X (v |u). So MSE = E [(Y − g (X ))2 ] ∞ ∞ (v − g (u))2 fY |X (v |u)dv fX (u)du. = −∞ (4.26) −∞ For each u fixed, the integral in parentheses in (4.26) has the same form as the integral (4.25). Therefore, for each u, the integral in parentheses in (4.26) is minimized by...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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