Body weights

Assume that body weights of men are Normally distributed with a mean of 170 pounds and a standard deviation of 30 pounds.

What is the body weight threshold separating the lightest 90% from the heaviest 10%?

Potholes spacing

The distance (, measured in miles) between successive potholes on a highway follows an exponential distribution with parameter per mile.

For a research project, a civil engineer is interested in learning more about the square roots of the distances between successive potholes. What is the PDF of ?

Series of games, joint PMF

Two teams, A and B, will play a series of up to 3 games. The series ends when either team wins 2 games, so there are only 2 games if either team wins the first two games. A game can’t end in a tie. The outcomes of the games are independent of each other.

The probability of team A winning any game is 0.6.

Let X denote the total number of games in the series. Let Y denote the number of games won by team A .

(a) Construct the joint PMF of and .

(b) Given that 3 games were played, find the minimum mean square error estimate of the number of games won by A.

Comparator circuit

I have designed a comparator circuit with a multiplexer that chooses the smaller of two inputs and .

Suppose and are independent uniform continuous random variables with range .

Find the PDF of the output . (Include bounds for .)

Adder circuit

I have designed an analog adder circuit that adds two inputs and . So the output is . Assume and are independent uniform continuous random variables with range .

Find the CDF of the output .

Community college ages

At a community college, the mean age of the students is 22.3 years, and the standard deviation is 4 years. A random sample of 64 students is drawn.

(a) What is the probability that the average age of the students in the random sample is less than 23 years?

(b) Use Markov’s Inequality to find an upper bound for the probability that the average age of the students in the random sample is more than 23 years.

(c) What is the probability that the total age of the students in the random sample is less than 1472 years?

Doorbell system photons. Otherwise, when a person is present (second hypothesis), the photodetector output is a Poisson RV with expected value 3000 photons.

An automatic doorbell system rings a bell whenever it detects someone by the door. The system uses a photodetector. If no one is present (null hypothesis), the photodetector output N is a Poisson RV with expected value

Assume 50% prior probability that someone is outside the door.

Design a binary hypothesis test to determine whether someone is outside the door. Clearly identify the two acceptance sets as a function of the number of photons.

Telemetry

A telemetry signal transmitted from a temperature sensor on a communications satellite is a random variable with and .

The receiver at mission control receives the signal , where is a noise voltage that is independent of and has this PDF:

Misplaced &0 & \text {otherwise}\end{cases}$$ Find the optimal linear estimator of $X$ for a particular value of $Y$.