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 ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co 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 ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co 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)