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

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 6Def 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

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