W15 Homework B

Due date: Tuesday 4/28, 11:59pm

Mean square error

01 $★$

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 $Y=5$.

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02 $★$

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 $Y=0.7$?

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03

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?

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Review

04

06

Bits received in error

In a digital communication channel, it is assumed that a bit is received in error with probability $4.3\times {10}^{-5}$. Someone challenges this hypothesis: they believe the error rate is higher than $4.3\times {10}^{-5}$. Assume 100,000 bits are transmitted. Design a one-tailed significance test using $\alpha =0.05$ and $N$, the number of bits received in error, to decide whether to reject the hypothesis that the error rate is $4.3\times {10}^{-5}$. Your rejection region should be of the form $\{N\ge c\}$. You do not have to use the continuity correction.

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Solution

Solutions---5300-06

05

05

CAT scan for tumors

When a brain is scanned in a CAT scan, analysis of the results yields a rating of 1, 2, 3, or 4. This represents (imperfect) evidence of whether there is a tumor.

	$X$	1	2	3	4
No tumor:	$P_X(k)$	0.4	0.3	0.2	0.1
	$X$	1	2	3	4
With tumor:	$P_X(k)$	0.0	0.1	0.3	0.6

Suppose that, of people who get CAT scans, 20% do have a tumor.

Furthermore, assume that declaring there is no tumor when there is one is ten times worse than declaring there is a tumor when there isn’t one.

Design an MC test to determine which ratings should be classified as tumors.

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