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**There are multiple question in this assignment. and i am skeptical about questions 2, 3 and 5 in the PDF. i need some explanations and answers on each section. I have attached the PDF file of the assignment. Can anyone go through and let me know on the answers of it with explanation. It would be helpful. **

ECE 3530: Spring 13 University of Utah HOMEWORK #4 - DUE: Friday, Feb 22

ECE 3530: Spring 13 University of Utah HOMEWORK #4 - DUE: Friday, Feb 22

Write your name on everything you hand in. Show your work.

1. Exercise 3.1 from textbook.

2. (a) The function

( f (x) = 1.5 ? 2|x|, ?1 ? x ? 1

0,

otherwise is not a valid probability density function. Why not?

(b) The function F (x) = 1 + sin(x) is not a valid cumulative distribution function.

Why not? sh is

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m (c) The function ( F (x) = 2 ? x1 ,

x>1

0,

otherwise is not a valid cumulative distribution function. Why not?

(d) Is the following valid probability mass function for a discrete random variable? If

it is not, state the reason. 0.2,

x=0

0.6,

x=1

f (x) = 0.2,

x=2 0,

otherwise (e) Consider the following probability mass function for a discrete random variable: 2k,

x = ?1

0.5,

x=0

f (x) = 3k,

x=1 0,

otherwise Find the value k which makes f (x) a valid probability mass function. Th 3. An electronics company manufactures 3 models of microprocessors. The models sell for

the following prices: Model A: $20, Model B: $40 and Model C: $60.

Customers buying microprocessors from this company choose the various models with

the following probabilities:

P (A) = 0.2 P (B) = 0.7 P (C) = 0.1 (a) A market analyst wishes to calculate the average price of chips that customers

purchase. Calculate this average or mean price.

(b) The plot below shows the probability density function f (x) for the maximum sustainable clock speed of the model A microprocessor. https://www.coursehero.com/file/12189195/ECE3530Spring13Assignment4/ sh is

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m Find the mean of the maximum sustainable clock speed for the model A microprocessor.

(c) Find the cumulative distribution function F (x) for this random variable.

(d) Compute the probability that the maximum sustainable clock speed for a model A

will be less than 60 kHz.

(e) Compute the probability that the maximum sustainable clock speed for a model A

will be between 60 and 80 kHz.

4. Exercise 3.36 from textbook. 5. A cellphone service provider wants to analyze the signal strengths of its network.

(a) Let X be the random variable that is the distance (in miles) that a customer will

be to the nearest cellphone tower. Analyzing their database, the company finds

that the probability density function is

( f (x) = 1/4, 2 ? x ? 6

0,

otherwise Find the mean and variance of random variable X.

(b) The company is now interested in signal strengths. The company finds that the

signal power (in milliwatts) at a distance x from the tower is P (x) = 40/x2 . Given

the probability density function for X in the previous part, what is the expected

signal strength? Hint: This is asking for E[P (x)] Th 6. A bag contains 7 red and 3 blue marbles. You draw three marbles from this bag without

replacement. Let X be the number of blue marbles you get.

(a) Is X a discrete or a continuous random variable?

(b) Determine the probability distribution function f (x) and the cumulative distribution function F (x) for the random variable X.

(c) Using F (x) find P (X ? 1)

(d) Compute the mean, variance and standard deviation of the random variable X.

(e) Assume that this is a game in a casino: the payoff is $100X and it costs $10, 000 to

play the game each time. Compute the average net gain/loss per game. https://www.coursehero.com/file/12189195/ECE3530Spring13Assignment4/ Powered by TCPDF (www.tcpdf.org)

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