**HMC Math 62: Probability & Statistics (Summer, 2002)**

Homework

**
FINAL: Due Saturday, June 29
**

### Assignment #14 (due Fri. 6/28):

**Chapter 12:** 1, 2, 4, 7abc

### Assignment #13 (due Thu. 6/27):

**Chapter 11:** 1, 2, 3a, 7, 15

### Assignment #12 (due Wed. 6/26):

**Chapter 9:** 8, 11

**Chapter 10:** 3, 5ac, 6, 9

### Assignment #11 (due Tue. 6/25):

**Chapter 9:** 4, 6abc, 12, 25

**Chapter 10:** 5ac, 6, 9

### Assignment #10 (due Mon. 6/24):

**Chapter 8:** 20, 22, 23, 28, 34, 55, 56

**
QUIZ II: Due Friday, June 21, 11:59pm
**

### Assignment #9 (less filling!) (due Fri. 6/21):

**Chapter 7:** 19, 23, 26, 27

**9A:** Suppose that X1, X2 ~ N(20,4) and X3, X4, X5 ~N(21,3.5), and all the Xi are independent. Define a new r.v. by Y=(X1+X2)/2 + (X3+X4+X5)/3. Compute P(Y>0) and P(-1 < Y < 1).

### Assignment #8 (due Thu. 6/20):

**Chapter 6:** 2a, 3abd, 5, 7, 11bc, 12bc, 13, 16, 18, 46

**8A:** Referring to problem 6.46a, compute this probability exactly; that is, find P(X >= 17) when X~Bin(30, 0.5). Recall that I showed you how to compute just this sort of probability using MATLAB. The command

`> sum(P(1,18:31))`

will compute the sum of entries 18 thru 31 in the row vector P. You may use something besides MATLAB to compute this, but it'd be good practice.

**8B:** WARNING: Thought required!

In class today, we saw how, given the pdf of r.v. X, we could compute the pdf of a new r.v. Y=aX+b. Use this method below.

Let r.v. X have pdf f(x)=x/2 , for 0 < x < 2. Define Y=aX+b and Z=X^2-2X ("X squared minus two X"). Find the pdf's for Y and Z. Then for each, verify that you've found a legal pdf by integrating it over the new variable's range of values. You should, of course, get 1.

### Assignment #7 (due Wed. 6/19):

Do Problems **7A** and **7B** in this GIF.

### Assignment #6 (due Tue. 6/18):

As our text sucks, I've had to steal problems from someone else's. So do Problems **6A-6E** in this GIF.

NOTES: Problems **7A** and **7B** for tomorrow are also included in this GIF.

Also, note that problem **6D**ef requires knowledge of the variance of a continuous random variable, which we have not yet discussed. It works just like it does for a discrete random variable. You can figure it out.

**6F:** Revisit the *E. coli* model in Problem 5.39. Recall that this situation was modeled with a Poisson distribution with parameter 2.5. Recall also our discussion in class regarding the relation between the Poisson and Exponential random variables.

**(i)** Use the Poisson distribution to find the probability that, in our size 100,000 population, no cases are reported this year.

**(ii)** Now use the Exponential distribution to compute this same probability. Do your answers agree?

**(iii)** Based on our qualitative description, if X~Expo(a), take a guess at the value of E[X]. No, wait... better yet, compute it precisely! Yeah! (You all remember integration by parts, don't you?) What, then, is our expected waiting time until the next *E. coli* outbreak?

### Assignment #5 (due Mon. 6/17):

**Chapter 4:** 90, 93

**Chapter 5:** 26, 29, 39

**5A:** Prove that if X~Bin(n,p) then Var(X) = np(1-p).

*HINT*: Use our proof for E[X] of a binomial as a model, and apply our formula about Var(X1+...+Xn).

**5B:** Suppose that a particle starts at the origin of the real line and moves along the line in jumps of one unit. For each jump the probability is p that it jumps left and 1-p that it jumps right. Find the expected position of the partical after n jumps. What is the variance of this position?

**5C:** Suppose a fair coin is tossed repeatedly until exactly k heads are observed. Let X be the number of tosses required. Find the probability mass function of X (i.e. P(X=k) for all appropriate k), and then find E[X].

*HINT FOR E[X]:* Let Xi be the number of tosses needed to get the i-th head after the (i-1)st head is observed.

** QUIZ I: Due Friday, June 14 **

### Assignment #4 (fewer calories!) (due Fri. 6/14):

**Chapter 4:** 84, 88, 89

**Chapter 5:** 18, 22, 32

NOTE: For 5.32, you can try to look up (a) in the Appendix or just work it out on your calculator.

### Assignment #3 (due Thu. 6/13):

**Chapter 4:** 49, 50, 51, 52, 63cdeh, 65, 73, 74

**3A:** Write down the "Binomial Theorem." I don't care where you find it... look it up, ask someone, whatever. Just write it down. Then, recall from class today we computed P(seeing k heads in n flips) for a biased coin with heads probability p. Use the Binomial Theorem to show that if I sum this quantity over k=0 to n, that I get one. Briefly explain (a sentence or two), why this is the expected result.

**3B:** An infinite geometric sum is 1 + x + x^2 + x^3 + ...., and is equal 1/(1-x) when |x|<1. Use this to show that if I sum P(1st heads appears on k-th flip) over all k, I get one. Again, why is this the expected result?

(NOTE: x^n means "x raised to the n-th power")

### Assignment #2 (due Wed. 6/12):

**Chapter 4:** 25, 29, 31, 39, 45c-i, 47, 48, 58

**2A:** Consider this diagram of events A, B, and C, with probabilities for each region indicated....

**(i)** Show that P(AnBnC) = P(A)*P(B)*P(C). **(ii)** Show that A & B are not independent events, and that B & C are not independent events. **(iii)** Are A & C independent?

This problem partially addresses a question raised in class today. Soon (maybe tomorrow) we'll see that three events may not be independent even if they are *pair-wise* independent.

**2B:** Using our three Axioms of Probability from class, prove that for any events A and B, P(AuB) = P(A)+P(B)-P(AnB). (Drawing and explaining Venn diagrams will *help*, but will not be considered a complete proof.)

### Assignment #1 (due Tue. 6/11):

**Chapter 4:** 4, 7, 8, 10, 14, 15, 24

**Also, Read (or skim) Chapers 1 & 2 as necessary, and do:**

**Chapter 2:** 5, 14, 21

**Return to: Math 62 Main Page * Greg Levin's Page * Department of Mathematics**

Email: *levin@hmc.edu*