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Linear Algebra Final Practice Problems - Set Notation Focus, Exams of Mathematics

Brown UniversityMathematics

This document contains a set of short-form practice problems for a linear algebra final exam. The problems emphasize the use of set notation and cover various topics typically found in a linear algebra course, such as vector spaces, linear transformations, and matrix operations. Ideal for students looking to review key concepts and sharpen their skills before the final exam. Created by [Author Name], Year [e.g., Fall 2024], for [Professor Name or Course Number].

Typology: Exams

2023/2024

Uploaded on 09/23/2024

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Final Exam Practice Problems APMA 1650
This practice set is an amalgamation of former final exam questions offered over the last several
semesters. Use this worksheet as an example of the types of questions and level of difficulty that
you can expect on the exam.
Note that this worksheet emphasizes the material post-midterm 2, though all course content will
be featured on the final exam. Don’t forget to study our midterm 1 and midterm 2 practice
material as well.
1. There are three printers in the Applied Math department and none of them works well.
The first printer has a 10% probability of jamming, the second has a 20% probability of
jamming, and the third has a 30% probability of jamming. Unfortunately, nobody knows
which is which.
(a) You send a printing job that will go to one of the printers at random. What is the
probability that you will encounter a paper jam?
(b) You send a printing job to a random printer and encounter a paper jam. What is the
probability that you picked the least reliable printer? What is the probability that you
picked the most reliable printer?
(c) If you send another printing job to the same printer that jammed in part (b), what is
the probability that it will jam again?
2. Let Xand Ybe two random variables with joint density function
f(x, y) = (4xy , 0≤x≤1,0≤y≤1
0,elsewhere
(a) What is the probability that X+Y < 1?
(b) Find the marginal density function for Y.
3. You survey 100 Brown students and ask whether or not they think all undergrads should
be required to take statistics. 30 of them say yes. Provide a 95% confidence interval for the
true proportion of the student body that would say yes.
4. It is known that 10% of people have green eyes. Say we select 7 people at random and
photograph them one by one.
(a) What is the probability that the first green eyed person is fourth in line?
(b) What is the probability that we have at most 2 green eyed people in the group?
(c) What is the probability that most of the people in the group have green eyes?
(d) If we photograph a group of 7 people every day, how many days would we expect to
need until we find a group that includes at least one green eyed person?
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Download Linear Algebra Final Practice Problems - Set Notation Focus and more Exams Mathematics in PDF only on Docsity!

This practice set is an amalgamation of former final exam questions offered over the last several semesters. Use this worksheet as an example of the types of questions and level of difficulty that you can expect on the exam.

Note that this worksheet emphasizes the material post-midterm 2, though all course content will be featured on the final exam. Don’t forget to study our midterm 1 and midterm 2 practice material as well.

  1. There are three printers in the Applied Math department and none of them works well. The first printer has a 10% probability of jamming, the second has a 20% probability of jamming, and the third has a 30% probability of jamming. Unfortunately, nobody knows which is which.

(a) You send a printing job that will go to one of the printers at random. What is the probability that you will encounter a paper jam? (b) You send a printing job to a random printer and encounter a paper jam. What is the probability that you picked the least reliable printer? What is the probability that you picked the most reliable printer? (c) If you send another printing job to the same printer that jammed in part (b), what is the probability that it will jam again?

  1. Let X and Y be two random variables with joint density function

f (x, y) =

4 xy , 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 0 , elsewhere

(a) What is the probability that X + Y < 1? (b) Find the marginal density function for Y.

  1. You survey 100 Brown students and ask whether or not they think all undergrads should be required to take statistics. 30 of them say yes. Provide a 95% confidence interval for the true proportion of the student body that would say yes.
  2. It is known that 10% of people have green eyes. Say we select 7 people at random and photograph them one by one.

(a) What is the probability that the first green eyed person is fourth in line? (b) What is the probability that we have at most 2 green eyed people in the group? (c) What is the probability that most of the people in the group have green eyes? (d) If we photograph a group of 7 people every day, how many days would we expect to need until we find a group that includes at least one green eyed person?

  1. The times that a cashier spends processing each customer’s order are independent random variables with mean 2.5 minutes and standard deviation 2 minutes. What is the probability that it will take more than 4 hours to process the orders of 100 people?
  2. The output voltage for an electric circuit is specified to be 130 with standard deviation 4. A sample of 64 independent readings on the voltage for this circuit gave a sample mean 128.6. Test the hypothesis that the average output voltage is 130 against the alternative that it is less than 130. Use a test with level .01.
  3. Let X 1 , X 2 ,... , Xn be a random sample of size n > 2 from a distribution with mean μ > 0.

(a) Let θ 1 =

X 1 + Xn 2

Compute the bias of θ 1 as an estimator for μ. Is θ 1 an unbiased estimator? (b) Let

θ 2 =

X 1 +···+Xn− 1 n− 1 +^ Xn n

Is θ 2 an unbiased estimator for μ? (c) Let θ 3 = θ 1 + kθ 2. Find the value of k for which θ 3 is an unbiased estimator for μ.

  1. The Public Health Department gives us the following information for a new diagnostic test. The test yields a positive result 90% of the time when the disease is present, while the test yields a positive result 1% of the time when the disease is not present. The actual number of individuals with the disease is unknown. Instead, let n represent the number of individuals that have the disease per 1,000 people. In terms of n, what is the probability that an individual who tests positive for the disease actually has the disease?
  2. Let X, Y be two discrete random variables with joint pmf

p(x, y) =

xy^2 15

, x = 1, 2 , y = 0, 1 , 2.

(a) Find the marginal pmfs pX (x) and pY (y). (b) Are X and Y independent? Why or why not? (c) Determine the means μX and μY. (d) Determine the covariance Cov(X, Y ).

  1. The administrators for a hospital wish to estimate the average number of days required for inpatient treatment of patients between the ages of 25 and 34. A random sample of 500 hospital patients between these ages produced a mean equal to 5.4. Construct a 95% confidence interval for the mean length of stay for the population of patients from which the sample was drawn given that the population standard deviation is 3.1 days.
  1. Let Y 1 , Y 2 ,... , Yn denote a random sample from the probability density function

f (x, θ) = (θ + 1)xθ, for 0 < x < 1 and θ > − 1.

Find the MLE for θ. Recall that (^) dxd (ax) = ax^ · ln a, for any positive real number a.

  1. A forester studying the effects of fertilization on certain pine forests in the Southeast is interested in estimating the average basal area of pine trees. In studying basal areas of similar trees for many years, he has discovered that these measurements (in square inches) are normally distributed with standard deviation 4 square inches.

(a) If the forester samples n = 9 trees, find the probability that the sample mean will be more than 2 square inches above the population mean. (b) Suppose the forester would like the sample mean to be within 1 square inch of the population mean, with probability .90. How many trees must he measure in order to ensure this degree of accuracy?