Question Details

[solution] » How do I do number 9? Can someone walk me through this problem

Brief item decscription

Step-by-step solution file

Item details:

How do I do number 9? Can someone walk me through this problem

How do I do number 9? Can someone walk me through this problem

36-225 ? Introduction to Probability Theory ? Fall 2016


Homework 5


Due Friday, October 7th at Noon in the mailbox outside the BH 132 doorway


Exercise numbers refer to Wackerly 7. If you are not using the American print edition of Wackerly 7:


check the problem scans on Blackboard to ensure you are completing the correct exercise!


Feel free to use R functions such as, e.g., dpois() to compute probabilities. Simply write down on your


homework sheet the R function call, and be sure to include function call arguments. However, realize that


you cannot use R on a test, so make sure you know how to compute any given probability by hand! (Thus it is


best to use R to confirm your by-hand computations of probabilities, and not to replace such computations.)


Test 1 will cover the hypergeometric and Poisson distributions, but not the uniform or normal


distributions. Keep that in mind when reviewing these homework problems before the October


11th exam.


1) [10 pts] In your pocket is a random number of coins N , where N ? Poisson(?). You toss each coin once,


with heads showing with probability p each time. Show that the total number of heads has the Poisson


distribution with parameter ?p.1


2) [7 pts] Exercise 3.112


3) [7 pts] Exercise 3.118


4) [7 pts] Exercise 3.124. Use the R function ppois() to estimate ?; never use a table.


5) [7 pts] Exercise 3.128


6) [7 pts] Exercise 3.138


7) [10 pts] Exercise 3.140


8) [7 pts] Exercise 4.40


9) [7 pts] A r.v. Y is sampled from a piecewise constant distribution: 1/3 0 ? y < 1


2/3 1 ? y < 2 .


f (y) = 0




(a) Compute E[Y ]. (b) Compute P (Y > 1|Y < 1.5).


10) [10 pts] A binary message is transmitted as a signal S, which we?ll say can take on the values ?1 or


+1 with equal probability. The transmission channel corrupts the message with additive noise, i.e., the


received value R = S + , where  ? N (0, ? 2 ). The receiver assumes S = ?1 if R < 0 and assumes


S = +1 if R ? 0. What is the probability of the receiver making an error (i.e., what is the probability of


S and R having opposite sign)? Assume S and  are independent random variables. State your answer


in terms of standard normal cdf function ?().








11) [7 pts] Show analytically that ?? e?x dx = ?, and thus show that






(x ? µ)2


f (x) = ?


exp ?


2? 2














is a density function if ? > 0. (Hint: note that ?? e?x dx =


e?x2 dx ?? e?y2 dy, which recasts




the original integral into a double integral that you may find easier to work with . . . particularly if you


do a coordinate transformation.)2


1 This


2 This is based on Problem 3.5.2 of Grimmett & Stirzaker, Probability and Random Processes, 2nd edition.


is based on Problem 4.14.1 of Grimmett & Stirzaker, Probability and Random Processes, 2nd edition. NOTE: In problems 12-13, you will need to use R or some other calculating device to evaluate your answer.


Do not use a probability table such as the one on the inside of the front cover of Wackerly 7! No one uses


tables nowadays.


12) [7 pts] A manufacturer creates widgets whose mass has mean 5 and standard deviation ?. Ten widgets


are shipped, and the shipment will be rejected if any of the widgets has mass ? 4 or ? 6. Give the


largest value ? can be such that the probability of rejecting the shipment is ? 0.01.


13) [7 pts] Exercise 4.74 (but not part e) Be sure:


? To make this work your own even if you work with others! Do not blindly copy the solutions; this is


plagarism, and furthermore it will only hurt you at exam time.


? To put your name on all sheets, and to staple the sheets together, if you are turning in multiple sheets.


? To properly label all plots (axes, main title, units, etc.) for full credit, if applicable, and to include a


printout of any code used in solutions, if applicable.


About this question:

This question was answered on: Feb 21, 2020

PRICE: $24 (18.37 KB)

Buy this answer for only: $24

This attachment is locked

We have a ready expert answer for this paper which you can use for in-depth understanding, research editing or paraphrasing. You can buy it or order for a fresh, original and plagiarism-free copy (Deadline assured. Flexible pricing. TurnItIn Report provided)

Pay using PayPal (No PayPal account Required) or your credit card. All your purchases are securely protected by PayPal.

Need a similar solution fast, written anew from scratch? Place your own custom order

We have top-notch tutors who can help you with your essay at a reasonable cost and then you can simply use that essay as a template to build your own arguments. This we believe is a better way of understanding a problem and makes use of the efficiency of time of the student. New solution orders are original solutions and precise to your writing instruction requirements. Place a New Order using the button below.

Order Now