ECE 109 UCSD Compute the Variance Using Gaussian Random Variable Questions

User Generated

Fhcxvqf6

Engineering

ece 109

University of California San Diego

ECE

Description

finish question 4, 5, 6

clear work

show all work step by step

Unformatted Attachment Preview

University of California, San Diego Department of Electrical and Computer Engineering ECE 109: Problem Set #5 1. Suppose that the continuous random variable X has probability density function f X (u) defined as follows: f X (u) = c(1 − u)2 for 0 < u < 1, and f X (u) = 0 elsewhere. (a) Find the value of the constant c. (b) Find the mean and the variance of X. (c) Compute P(6X 2 > 5X + 1) and P(6X 2 > 7X − 2). 2. Suppose that X is a geometric random variable with parameter p = 0.5, and let Y = sin (π X /2). Is Y discrete, continuous, or mixed? Find the CDF and either the p.d.f. or the p.m.f. of Y. 3. A box contains 4 white balls and 4 black balls, and the following game is played. Each round of the game consists of randomly selecting four balls from the box. If exactly two of them are white, the game is over. Otherwise, the four balls are placed back in the box, so that it again contains 4 white balls and 4 black balls before the next round is performed. The rounds are repeated until exactly two of the chosen balls are white. (a) Let X denote the number of rounds in the game. Find the expected value and variance of X. (b) When the game is over, we get a reward Y. The value of Y decreases exponentially with the the number of rounds played. Specifically, if the game terminates after n rounds, then Y = 1/en . Find the expected value of the reward Y. 4. Let X denote a Poisson random variable with parameter λ = 0.5. (a) Compute E[ X!]. (b) Show that E[ X n ] = 0.5 E[( X + 1)n−1 ]. (c) Use this result to compute E[ X 3 ]. 5. The width of a screw produced by a steel company is a Gaussian random variable with mean µ = 0.9 cm and standard deviation σ = 0.003 cm. (a) If the width specification limits of a customer are 0.9 ± 0.005 cm, and a screw whose width is not within these limits is deemed defective, what percentage of the screws produced by the company will be defective? (b) The CEO’s goal is to make sure that, on the average, no more than 1 in 100 screws produced are defective. What is the maximum allowable value of σ that will enable the company to achieved this goal? (c) Assume that µ = 0.9 cm and σ = 0.003 cm as before, but the customer changes specification limits to a minimum of 0.896 cm and a maximum 0.908 cm. What percentage of the screws produced will be defective now? 6. Let X be a Gaussian random variable with mean µ = −1 and variance σ 2 = 4. (a) Find the mean and variance of 2X + 5 Let Φ( x) denote the CDF of a standard Gaussian random variable, and let Q( x) = 1 − Φ( x). Suppose that Calculator A can evaluate only Φ( x) and only for nonnegative values of x. On the other hand, suppose that Calculator B can evaluate only Q( x), again only for x ≥ 0. Both calculators can perform standard functions, like addition and multiplication. For each of the probabilities in parts (b) through (e), write down two alternative expressions: one for evaluation using Calculator A, and the other for evaluation using Calculator B. (b) P( X < 0) (c) P(−10 < X < 5) (d) P(| X | ≥ 5) (e) P( X 2 − 3X + 2 < 0)
Purchase answer to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Explanation & Answer

Hello, am done with Q4, 5 and 6 as agreed. I have attached the two sources that I used to complete the work. The images are clearly written as agreed.

Running head: ENGINEERING QUESTIONS

Student Name
Institution
C...


Anonymous
Really helpful material, saved me a great deal of time.

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Related Tags