Isye 2027

# Their joint pdf is the product of their individual

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online