First Midterm Exam
APMA 3100
21 Feb 2025
Print name: ______________________________
Computing ID: ______________________________
Section: ______________________________
- You have 50 minutes to complete the exam.
- No books, personal paper, computers, or visible phones. Scientific calculators are allowed.
- You must deposit your phone with the instructor while using the bathroom if you wish to continue the exam on your return.
- You may have water, snacks, pens, pencils, erasers.
- You are not allowed to communicate with anyone except the instructor during the exam.
- Please draw a box around your final answer for each subquestion.
- The structure, quality, and clarity of your entire solution will be assessed. The final answer alone may be worth only part of the problem value.
Honor pledge
“On my honor as a student, I pledge that I have neither given nor received aid on this exam.”
Sign your name, pledging your honor: ______________________________
Question 01 - Probability logic of events
On a random day in Virginia, the weather might rain, snow, or both, or neither.
The odds of rain are 40%. The odds of snow are 50%. The odds of neither are 20%.
(a) State the Inclusion-Exclusion principle for (any) two events
Question 02 - Conditioning and cases
(a) State Bayes’ Theorem.
Bin 1 holds 4 red and 3 blue marbles. Bin 2 holds 3 red and 2 blue marbles.
You take a random marble from Bin 1 and put it in Bin 2 and shake Bin 2.
Then you draw a marble from Bin 2. It is red.
(b) What is the probability that the marble transferred was also red? (No need to simplify your answer.)
Question 03 - Expectation and variance properties
(a) State the defining formula for variance
Question 04 - Poisson process
The Emergency Department gets 911 calls according to a Poisson process at an average rate of 6 calls per hour.
Write two formulas for the probability that the ED receives no call in the first two hours of the day, as follows:
(a) Write an appropriate Poisson variable and its PMF. (b) Use your Poisson PMF to solve the problem. (No need to integrate, compute summation, or simplify.) (c) Write an appropriate exponential variable and its PDF. (d) Use your exponential PDF to solve the problem. (No need to integrate, compute summation, or simplify.)
Question 05 - CDF and PDF of derived
Consider the following PDF for a random variable
Question 06 - Normal distribution
(a) Assume that
(b) A random variable