HMC Math 62: Probability & Statistics (Summer, 2001)
Homework

### Assignment #13 (due Fri. 6/22):

Section 10.1: 1, 2, 5, 10
Section 12.1: 7
Section 12.2: 19, 22

### Assignment #12 (due Thu. 6/21):

Section 8.1: 3, 5, 9, 12, 13
Section 8.2: 15
12A: Prove the Proposition on page 408. The proof is described in the following paragraph; present a more formal, algebraic write-up of it.

### Assignment #11 (due Wed. 6/20):

Section 7.1: 1, 2, 4ace, 7, 8
Section 7.2: 13, 23
Section 7.3: 32 (Use t-distribution and table on pg 725)

### Assignment #10 (due Tue. 6/19):

Section 6.1: 1abcd, 3abcd, 4, 9, 13, 16
Section 6.2: 20
Read This (or at least some of it)

### Assignment #9 (due Mon. 6/18):

Section 5.4: 46, 47, 56
Section 5.5: 60, 65

### Assignment #8 (due Thu. 6/14):

Section 4.2: 19 (note: "cdf"=distribution, not density)
Section 4.3: 26eh, 27ace, 30acd, 33, 37, 39, 45
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.
8A: Suppose X~Expo(2). Use the above to find the pdf of Y=X^3 ("X cubed").
8B: 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 #7 (due Tue. 6/13):

Section 4.1: 5, 7, 10
Section 4.2: 15, 21, 23, 24
Section 4.4: 59, 61

### Assignment #6 (due Tue. 6/12):

Section 3.3: 28, 31, 32, 36, 40
Section 3.4: 50, 55, 56 (note: X~Bin(n,p)), 59
6A: Prove that if X~Bin(n,p) then Var(X) = np(1-p).
Hint: Use our proof from Friday for E[X] of a binomial as a model,
and apply our formula from today about Var(X1+...+Xn).
Solutions to HW#6

### Assignment #5 (due Mon. 6/11):

Section 3.3: 29, 34, 37
Section 3.4: 48, 51, 57a, 58 (do it based on the hint!)
Section 3.6: 76, 80

### Assignment #4 (due Fri. 6/8):

Section 2.4: 59, 63, 66
Section 3.1: 2, 5, 10
Section 3.2: 13, 16acd, 25
Section 3.4: 47

### Assignment #3 (due Thu. 6/7):

Section 2.4: 45, 51, 56, 57, 58
Section 2.5: 74, 76, 78, 80
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.

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

Section 2.1: 8
Section 2.2: 24, 26
Section 2.3: 34, 41, 42
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

### Assignment #1 (due Tue. 6/5):

Section 2.1: 1, 9
Section 2.2: 12, 17, 23
Section 2.3: 32, 38, 40, 43

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Email: levin@hmc.edu