**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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