Homework for Probability & Statistics
Damien Pitman
CHAPTER 0
Sampling, Data, and Descriptive Statistics
1. A study was done to determine the proportion of college students that work full
time. The sample proportion of college students that work full time in this study
is the:
A. statistic B. parameter C. population D. variable
2. Suppose that 4% of the population has red hair and that you are going to call
twenty five people selected at random from the phone book.
a. Explain why you would expect to have one person in your sample with red
hair.
b. Suppose your sample has 3 people with red hair. Do you think this sample came from a biased sampling technique or simply from random variation
among samples? Why?
3. Find the following statistics for the data given here.
sample size
sample mean
median
x f
mode is
1 1
minimum
2 2
maximum
3 2
first quartile
4 4
third quartile
interquartile range
sample standard deviation
4. Consider the data given in the last problem.
a. Make a box plot and a histogram for the given data.
b. Suppose that there was actually another data point, x = 7, collected in the
sample above, but that the researcher had chosen to throw it out because they
felt that it was an outlier. Comment on this decision and also on whether
the sample standard deviation would increase or decrease if this observation
were to be included. You do not need to calculate this alternate standard
deviation.
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CHAPTER 1
Basic Probability Theory
1. Draw one card from a standard deck of playing cards. For 1 ≤ i < 13, let Si be
the event that the card is drawn is the i or i + 1 of spades (where the face cards
Jack, Queen, and King are 11, 12, and 13 respectively).
a. What is the sample space for this experiment?
b. Describe S1 and S2 .
c. Find P (S1 ), P (S2 ), P (S1 ∪ S2 ), and P (S1 ∩ S2 ).
d. Find P (S c ) where S = ∪12
i=1 Si .
2. Consider Example 1.3.1. Explain the assumption that Mrs T must have been making when she said “that sounds more probable”.
3. Consider Example 1.3.3.
a. Why is the area corresponding to D ∩ F equal to 0.75?
b. Why is it not true that P ( even or odd ) = 1?
4. Consider Example 1.3.4. Use inclusion-exclusion to find the probability that the
chosen number is composite.
5. Use complements and a list of primes to double check your answer in the last
problem.
6. Suppose we pick a number uniformly at random from the interval [0, 2].
a. What is the sample space and what kind of infinite is it?
b. Check proposition 1.3.5 if An = [0, 1 + 1/n].
Ch. 1. 1, 4, 7, 8, 12
Due 9/15: All problems above since the last assignment
Ch. 1. 17(a,c,d), 18, 19, 20, 24, 27
Due 9/22: All problems above since the last assignment
1.33 Try some examples with specific probabilities assigned to A, B, and their conditionals.
1.35 Work from P (X ∩ Y ) = P (X|Y )P (Y ).
1.38 Think of how factoring relates to each of independence and being prime.
1.41 Just think about it and explain what you think. ... but be rigorous.
1.42 Just think about it and explain what you think. ... but be rigorous.
1.50 For (a), think about what A∩B∩C is. For (b), see if the definition of independence
for three sets is satisfied.
1.52 Let Bi = { dart i hits the bullseye}. Think about (a) and (b) in terms of the Bi .
1.53 Use the definition of conditional probability.
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1. BASIC PROBABILITY THEORY
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1.64 Look at the cancer example 1.6.2.
1.65 Make a tree and use the law of total probability.
1.68 Look at the HOUSTON example 1.6.1. Here you have to realize that you might
choose all the same, two the same, or none the same.
1.74 Look at example 1.6.4.
Due 9/29: All problems above since the last assignment
Note: The first exam will be over Chapter 1, and this homework completes our study
of Chapter 1.
Read Sections 2.1, 2.2, and 2.3.
2.3 Think about adding up probabilities. What is the total sum for (a)? What is the
sum up to and including 2 for (b)? Think complements for (c).
2.4 This is similar to 2.3.
2.5 Count the number of ways to get each number of aces and then remember to
divide by the total number of possible draws.
2.6 Find the pmf by considering all possibilities, then look at Fig. 2.4 for the cumulative distribution.
2.11 Think about the X ∼ unif[0,0.5].
2.13 Look at page 88.
2.15 Look at page 87.
2.17 For (a) and (b), look at pages 87-88. For (c), consider that
P (Y ≤ a) = P (1/X ≤ a) = P (X ≥ 1/a).
Use this to find the cdf for Y . Then look at page 87 for the relationship between
cdf’s and pdf’s.
2.18 This is similar to part (c) in 2.17, and we ultimately want to know what is the
distribution on the random variable Y = 1 − X.
Due 10/13: All problems above since the last assignment - and the reading
2.23 This is like problem 6 above - recall that we made a table of values for the random
variable an found the probability for each outcome associated to each value of the
random variable. Here, if we want the minimum, find how many ordered pairs
correspond to each value that the minimum could be. Same for the difference.
2.25 Find the probability associated to each of the possible number of aces, and then
do the extended table with kp(k).
2.26 Assume that the test is perfect. Then make a tree diagram to organize your thinking and to calculate probabilities associated to each possible outcome.
2.27 Find the probability associated to each of the possible number of matches, and
then do the extended table with kp(k).
2.62 This is like the exponential distribution problem done in class, but we have not
yet done the variance Var[T ]. You may want to look at Proposition 2.6.2 to find
the value of the variance.
2.63 Set up equations to solve for λ and see if they match.
1. BASIC PROBABILITY THEORY
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2.65 This is similar to the exponential distribution problem done in class, but you may
want to look at example 2.6.1.
Due 10/20: All problems above since the last assignment