W14 - Homework B

Due date: Thursday 12/04, 11:59pm

Mean square error

01

05

MMSE linear estimator from joint PMF

Suppose and have the following joint PMF:

			0
	0

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

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

(c) Use (a) to estimate given and .

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02

06

MMSE linear estimator from joint density

Consider this joint PDF:

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

(b) What is the linear estimate of given ?

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

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03

07

Telemetry signal

A telemetry signal, , transmitted from a temperature sensor on a communications satellite is a Gaussian random variable with and . The receiver at mission control receives , where is a noise voltage independent of with PDF:

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

(a) What is , the expected value of the received voltage?

(b) What is , the variance of the received voltage?

(c) What is , the covariance of the transmitted voltage and the received voltage?

(d) What is the correlation coefficient of and ?

(e) What are and , the optimum mean square values of and in the linear estimator?

(f) What is , 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 . Someone challenges this hypothesis: they believe the error rate is higher than . Assume 100,000 bits are transmitted. Design a one-tailed significance test using and , the number of bits received in error, to decide whether to reject the hypothesis that the error rate is . Your rejection region should be of the form . You do not have to use the continuity correction.

Solution

W14---HW-Solutions#04

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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.

		1	2	3	4
No tumor:		0.4	0.3	0.2	0.1
		1	2	3	4
With tumor:		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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