**HMC Math 62: Probability & Statistics (Summer, 2004)**

Homework

**
FINAL (Cover Sheet): Due Friday, June 25 (midnight)
**

(You may view this cover sheet at any time)

### Assignment #12 (due Thu. 6/24):

**12A:** Some problem I read in another prob/stat book said some damn thing about a linear regression model on concrete strength that didn't make any sense to me. But it had y = 28-day standard-cured strength (psi), and x = accelerated strength (psi), whatever that means. What's more, it would have us suppose that the equation of true regression is y = 1800 + 1.3x. It's not much, but it's enough info for a statistics homework problem, so we're all set!

*(i)* What is the expected 28-day strength when accelerated strength = 2500?

*(ii)* By how much can we expect 28-day strength to change when accelerated strength increases by 1 psi?

*(iii)* Answer *(ii)* for an increase of 100 psi.
**Chapter 14:** 1ad

**12B:** 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.

NOTE: If you're a big smarty-pants and have gotten perfect scores on all your homeworks, then you CAN'T GET THIS PROBLEM RIGHT! BWAAHAHAHAHA!!! You'll no longer have a perfect homework score! Bow your head in shame!

Ok, only kidding... since I'll be dropping your lowest two homework scores, I guess this would just be one of those. So it'll all be ok.

**12C:** Do **12B:** again for a problem from a different homework assignment.

### Assignment #11 (due Wed. 6/23):

**11C:** Use the data in problem 12-2 on page 350 of our text to do an analysis of variance test at a .05 significance level. Summarize your results in an ANOVA table.

**11D:** Today in class I claimed that MSTr and MSE are unbiased estimators of variance when then null hypothesis is true. That is, if the null hypothesis is true, then E[MSTr] = E[MSE] = "sigma squared". Prove these.

**HINTS**: For MSE, use the first formula given in class today, and what you know about the sample variance S^2. For MSTr, recall that, if the null hypothesis is true, then all sample means are i.i.d. normal with the same mean, and that the sample variance of *these* data points is an unbiased estimator for the variance of a sample mean. NOTE: I would *not* recommend trying to follow or reproduce the proof starting on page 325 of our text. The graders will not give credit for a proof with taus and epsilons.

### Assignment #10 (due Tue. 6/22):

First, do Problems **10A-10E** in this GIF. Then do:

**Chapter 11:** 3 (assume an underlying normal distribution)

Solutions

### Assignment #9 (due Mon. 6/21):

**Chapter 10:** 7, 37, 40, 42, 48abc, 66, 67

**NOTE on 10.48ab**: Assume that the distribution is normal. Because n is small and variance is unknown, you'll need to compute (a) and (b) using a t-distribution... see table on pg 604)

### Assignment #8 (low carb!) (due ~~Fri 6/18~~ Mon 6/21):

**Chapter 8:** 19, 20

**Chapter 10:** 1, 2, 20

**NOTE on 10.1 and 10.2**: When the problems ask "Which is the better estimator?", the correct answer is "The one with the smaller variance." To this end, find the variance of each estimator.

Read This
(or at least some of it)

**
QUIZ II (Cover Sheet): Due Friday, June 18
**

(You may view this cover sheet at any time)

### Assignment #7 (due Thu. 6/17):

**Chapter 7:** 1acde, 3ab, 5bc, 8, 14, 23, 24, 29

**7A:** The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter a=50. What is the approximate probability that

**(i)** Between 35 and 70 tickets are given out on a particular day? (Hint: When a is large, the Poi(a) distribution may be approximated by a normal distribution with appropriate mean and variance.)

**(ii)** The total number of tickets given out during a five day week is between 225 and 275? (Hint: If tickets given in one day is Poi(50), then tickets given in 5 days should also be Poisson... with what mean?)
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.

**7B:** Suppose X~Expo(2). Use the above to find the pdf of Y=X^3 ("X cubed").

**7C:** 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 Wed. 6/16):

**Chapter 2:** 21abd

Do Problems **6A-6D** in this GIF.
Solutions

### Assignment #5 (due Tue. 6/15):

**Chapter 2:** 4, 11ac

**Chapter 6:** 2, 6, 13, 16

**5A:** Refer to problem 5.30. Suppose that the system can only handle 16 calls. Determine the probability of an overload using the Poisson table on page 600 of your text.

**5B:** Revisit the tool crib model in Problem 5.34. Recall that this situation was modeled with a Poisson distribution with parameter 2 (orders/day). 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 tool crib receives no orders on a given day.

**(ii)** Now use the Exponential distribution to compute this same probability. Do your answers agree?

### Assignment #4 (due Mon. 6/14):

**Chapter 2:** 1, 2, 15ab

**Chapter 3:** 4

**Chapter 5:** 9, 11, 12, 34

**4A:** 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). Do *not* use the proof given in the book!

**4B:** 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.

**4C:** Consider problem 5.7 in the text. You are asked to find P(X<=3) where X~Bin(100,.01). To do this, you will need to compute 4 distinct Binomial terms. Find the value of each, then add them. Now recall from class my claim that Bin(n,p) could be approximated by Poi(np). Use this to find a Poisson approximation for each of the four Binomial terms , and then find the sum of your approximations. (Pretty good approximation, huh?)
Solutions

**
QUIZ I (Cover Sheet): Due Friday, June 11
**

(You may view this cover sheet at any time)

### Assignment #3 (due Thu. 6/10):

**Chapter 1:** 25, 30, 36, 38, 40

**3A:** A sports car comes with either automatic or manual transmission, and in one of four colors. Proportions for types sold are given below.
| White | Blue | Black | Red |

Automatic | .15 | .10 | .10 | .10 |

Manual | .15 | .05 | .15 | .20 |

Let A={automatic transmission} , B={black} , C={white}.
**(i)** Calculate P(A), P(B), and P(AnB)
**(ii)** Calculate both P(A|B) and P(B|A)
**(iii)** Calculate P(A|C) and P(A|C')

**3B:** For any events A and B with P(B) > 0, show that P(A|B) + P(A'|B) = 1.

**3C:** 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. 6/9):

**Chapter 1:** 4, 15, 22, 34(part 2 only)

**HINT on 34:** You'd be better off *not* using the book's proof as a model for this one. We haven't studied conditional probabilities yet, and it's just not necessary. Instead, as a first step, start with P(A')P(B') = (1-P(A))(1-P(B)) = ...

**Chapter 5:** 2
**2A:** An urn contains 8 red and 12 blue balls.

**(i)** How many ways are there to select 5 balls from the urn?

**(ii)** In how many ways can the sample of 5 balls contain exactly 3 red ones?

**(iii)** What is the probability that exactly 3 of the 5 balls will be red?

**2B:** Suppose that the proportions of blood phenotypes in a particular population are as follows:

Assuming that the phenotypes of two randomly selected indivduals are independent, what is the probability that both phenotypes are O? What is the probability that the phenotypes of two randomly selected individuals match?

**2C:** 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. Use Axioms II and III to explain why this is the expected result.
Solutions

### Assignment #1 (due Tue. 6/8):

**Chapter 1:** 2e, 3ce, 7, 12, 13, 14, 16, 18, 19
Solutions

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