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

36-225 ? Introduction to Probability Theory ? Fall 2016

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

otherwise

(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 ?().

R?

?

2

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

1

(x ? µ)2

f (x) = ?

exp ?

2? 2

2??

qR

R?

R?

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.

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