HMC Math 62: Probability & Statistics (Summer, 2003)
Homework

### Assignment #13 (due Thu. 6/26):

Section 12.1: 7
Section 12.2: 19, 23
13A: Find a problem from a previous homework that you missed and/or do not fully understand. Transcribe (or summarize) the problem statement, and write a sentence or three about what you don't understand. Be sure to bring this to class tomorrow for the exam review.
BONUS: Do 13A again for another problem from a different section.

### Assignment #12 (due Wed. 6/25):

Section 8.4: 47
Section 10.1: 1, 5, (9)
12A: Do problem 10.1.9, and also summarize your intermediate calculations in an ANOVA table like Table 10.2 (pg 411) and the one in example 10.3 (pg 412).
12B: Prove the first part of the Proposition on page 408, namely that
If the null hypothesis is true, then E[MSTr] = E[MSE] = "sigma squared".
The proof is described in the paragraph that follows; present a more formal, algebraic write-up of it.
Yes, this is the one I was looking at funny in class today... I'd forgotten that it's actually not hard at all. Well, relatively speaking.

### Assignment #11 (due Tue. 6/24):

Section 8.1: 5, 9, 12
Section 8.2: 15, 17, 19

QUIZ II (Cover Sheet): Due Monday, June 23
(You may view this cover sheet at any time)

### Assignment #10 (due Mon. 6/23):

Section 7.1: 1ac, 2, 5ad, 7
Section 7.2: 13, 23
Section 7.3: 37ac (Use t-distribution and table on pg 725)
Read This (or at least some of it)

### Assignment #9 (due Fri. 6/20):

Section 6.1: 1abcd, 3abcd, 9, 13, 16
Section 6.2: 20

### Assignment #8 (due Thu. 6/19):

Section 4.3: 37, 39
Section 5.4: 46, 47, 55, 56
Section 5.5: 59acd, 73

### Assignment #7 (due Wed. 6/18):

Section 4.2: 21, 23, 24
Section 4.3: 26eh, 27ace, 30acd
Section 4.4: 59
Recall our calculation in class today wherein, given normal X,
we found the density function of Y = g(X) = aX+b. It involved
(i) Finding the legal range of Y based on g and X.
(ii) Finding the cdf of Y in terms of the cdf of X.
(iii) Finding the pdf of Y by differentiating its cdf.
Although we applied this technique to a very simple function g,
it can be used for many functions.
7A: Suppose X~Expo(2). Use the above to find the pdf of Y=X^3 ("X cubed").
7B: Suppose X~U(0,1). Use the above to find the pdf of Y=e^X ("e to the X").
(NOTE: If you're so inclined, you can check your work be seeing if your pfd integrates to 1)

### Assignment #6 (due Tue. 6/17):

Section 3.4: 45deg
Section 3.6: 75abd
Section 4.1: 5, 7, 10
Section 4.2: 11, 15abcd
6A: Revisit the towing service model in Problem 3.81. Recall that this situation was modeled with a Poisson distribution with parameter 4 (calls/hour). 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 the towing service receives no calls this hour.
(ii) Now use the Exponential distribution to compute this same probability. Do your answers agree?

QUIZ I (Cover Sheet): Due Monday, June 16
(You may view this cover sheet at any time)

### Assignment #5 (due Mon. 6/16):

Section 3.3: 29, 30, 34, 37
Section 3.4: 55, 59
Section 3.6: 81
5A: Prove that if X~Bin(n,p) then Var(X) = np(1-p).
Hint: Use our proof from class for E[X] of a binomial as a model,
and apply our formula about Var(X1+...+Xn).
5B: Suppose a fair coin is tossed repeatedly until exactly n 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.

### Assignment #4 (due Fri. 6/13):

Section 3.1: 2, 5
Section 3.2: 13, 16acd, 17
Section 3.4: 44bc, 47, 51

### Assignment #3 (due Thu. 6/12):

Section 2.4: 45, 51, 56, 57, 63
Section 2.5: 79, 81
3A: 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?

### Assignment #2 (due Wed. 6/11):

Section 2.2: 17, 24
Section 2.3: 33, 41
Section 2.5: 71, 72
2A: Using Axioms I, II, III of probability, prove...
(i) P(AuB) = P(A) + P(B) - P(AnB)
(ii) P(AuB) <= P(A) + P(B)
NOTES: AuB is "A union B" ; AnB is "A intersect B"
These are proved in the text; try to work them out on your own
2B: 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, I get one. Briefly explain (a sentence or two), why this is the expected result.

### Assignment #1 (due Tue. 6/10):

Section 2.1: 1, 8, 9
Section 2.2: 12
Section 2.3: 29, 31, 38, 40, 43

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