Problems

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 $E[Y]$.)

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 $H=\{Y>3/2\}$.

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 $X=x$?

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 $X$?

05

MMSE linear estimator from joint PMF

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

$Y\downarrow X\to$	$-1$	$0$	$1$
$1$	$\frac{1}{6}$	$\frac{1}{12}$	0
$3$	$\frac{1}{8}$	$\frac{1}{12}$	$\frac{1}{24}$
$5$	$\frac{1}{24}$	$\frac{1}{12}$	$\frac{1}{8}$
$7$	0	$\frac{1}{12}$	$\frac{1}{6}$

(a) Find the minimal MSE linear estimator for $X$ in terms of $Y$.

(b) What is the MMSE error for this linear estimator?

(c) Use (a) to estimate $X$ given $Y=1$ and $Y=5$.

06

MMSE linear estimator from joint density

Consider this joint PDF:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}\frac{3}{2}(x^2+y^2)& x,y\in [\mathrm{0,1}]\\ 0& \text{ otherwise }\end{array}$$

(a) What is the minimal MSE linear estimator for $X$ in terms of $Y$?

(b) What is the linear estimate of $X$ given $Y=0.7$?

(You may use the data you found in W09-B Q8, you don’t need to repeat those calculations.)

07

Telemetry signal

A telemetry signal, $T$, transmitted from a temperature sensor on a communications satellite is a Gaussian random variable with $E[T]=0$ and $\mathrm{Var}[T]=9$. The receiver at mission control receives $R=T+X$, where $X$ is a noise voltage independent of $T$ with PDF:

$$f_X(x)=\{\begin{array}{cc}1/6& -3\le x\le 3\\ 0& \text{ otherwise }\end{array}$$

The receiver uses $R$ to calculate a linear estimate of the telemetry voltage:

$${\hat{T}}_L(R)=aR+b$$

(a) What is $E[R]$, the expected value of the received voltage?

(b) What is $\mathrm{Var}[R]$, the variance of the received voltage?

(c) What is $\mathrm{Cov}[T,R]$, the covariance of the transmitted voltage and the received voltage?

(d) What is the correlation coefficient ${\rho }_{T,R}$ of $T$ and $R$?

(e) What are $a^{\ast }$ and $b^{\ast }$, the optimum mean square values of $a$ and $b$ in the linear estimator?

(f) What is $e_L^{\ast }$, the minimum mean square error of the linear estimate?