(→Problem 1: Random Point, Revisited) |
(→Problem 1: Random Point, Revisited) |
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| − | *(a) | + | *(a) Find the marginal pdf <math>f_X(x)</math> of the random variable <math>X</math>. Find <math>E[X]</math> and <math>Var(X)</math>. |
| + | *(b) Using your answer from part (a), find the marginal pdf <math>f_Y(y)</math> of the random variable <math>Y</math>, and its mean and variance, <math>E[Y]</math>, and <math>Var[Y]</math>. | ||
| + | *(c) Find <math>f_{Y|X}(y|\alpha)</math>, the conditional pdf of <math>Y</math> given that <math>X = \alpha</math>, where <math>0 < \alpha < 1/2</math>. Then find the conditional mean and conditional variance of <math>Y</math> given that <math>X = \alpha</math>. | ||
| + | *(d) What is the MMSE estimator, <math>\hat{y}_{\rm MMSE}(x)</math>? | ||
| + | *(e) What is the Linear MMSE estimator, <math>\hat{y}_{\rm LMMSE}(x)</math>? | ||
== Problem 2: Variable Dependency== | == Problem 2: Variable Dependency== | ||
Revision as of 07:03, 2 December 2008
Contents
Instructions
Homework 10 can be downloaded here on the ECE 302 course website.
Problem 1: Random Point, Revisited
In the following problems, the random point (X , Y) is uniformly distributed on the shaded region shown.
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- (a) Find the marginal pdf $ f_X(x) $ of the random variable $ X $. Find $ E[X] $ and $ Var(X) $.
- (b) Using your answer from part (a), find the marginal pdf $ f_Y(y) $ of the random variable $ Y $, and its mean and variance, $ E[Y] $, and $ Var[Y] $.
- (c) Find $ f_{Y|X}(y|\alpha) $, the conditional pdf of $ Y $ given that $ X = \alpha $, where $ 0 < \alpha < 1/2 $. Then find the conditional mean and conditional variance of $ Y $ given that $ X = \alpha $.
- (d) What is the MMSE estimator, $ \hat{y}_{\rm MMSE}(x) $?
- (e) What is the Linear MMSE estimator, $ \hat{y}_{\rm LMMSE}(x) $?
