W15 Homework A

Due date: Thursday 4/23, 11:59pm

Mean square error

01

01

MSE of a derived variable

Suppose that $X\sim \mathrm{Unif}[\mathrm{0,2}]$, and $Y=X^2$.

You are using the sample mean with 50 samples, namely $M_{50}(Y)$, to estimate $E[Y]$.

What is the mean square error of $M_{50}(Y)$? (Errors are deviations of this variable from $E[Y]$.)

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02

02

Estimates from joint PDF

Suppose $X$ and $Y$ have the following joint PDF:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}{\displaystyle \frac{6(y-x)}{27}}& 0\le x\le y\le 3\\ 0& \text{otherwise}\end{array}$$

(a) Find $f_X(x)$ and the blind estimate ${\hat{x}}_B$.

(b) Compute ${\hat{x}}_G$, the MMSE estimate of $X$ assuming the event $G=\{X<3/2\}$.

(c) Find $f_Y(y)$ and the blind estimate ${\hat{y}}_B$.

(d) Compute ${\hat{y}}_H$, the MMSE estimate of $Y$ assuming the event $H=\{Y>3/2\}$.

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03

03

MMSE exact estimator from joint PDF

Suppose $X$ and $Y$ have the following joint PDF:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}2(y+x)& 0\le x\le y\le 1\\ 0& \text{otherwise}\end{array}$$

(a) What is ${\hat{x}}_M(y)$, the MMSE estimate of $X$ given $Y=y$?

(b) What is ${\hat{y}}_M(x)$, the MMSE estimate of $Y$ given $X=x$?

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04

04

MMSE linear estimator from joint PDF

Suppose $X$ and $Y$ have the following joint PDF:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}2(y+x)& 0\le x\le y\le 1\\ 0& \text{otherwise}\end{array}$$

(a) What is ${\hat{X}}_L(Y)$, the MMSE linear estimator of $X$ in terms of $Y$?

(b) What is ${\hat{Y}}_L(X)$, the MMSE linear estimator of $Y$ in terms of $X$?

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