**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)

### Midterm: Takehome, Due Monday, June 18

### 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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