Midterm II - APMA 3100 - McMillan - Spring 2025

Second Midterm Exam

APMA 3100 Sec. 003 & 004 McMillan

28 Mar 2025
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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 - Joint PMF

(a) Suppose is a fixed event written in discrete variables and .

Write a formula for :

Now suppose and have joint and marginal PMFs described by this table:

	0	1	2
1	0.05	0.07	0.08	0.20
2	0.08	0.10	0.12	0.30
3	0.12	0.33	0.05	0.50
	0.25	0.50	0.25

(b) Find the conditional PMF . Put your answer in the table:

	0	1	2
1
2
3

(c) Find . (d) Find the PMF of (use the original table).

(e) Compute . (Hint: Short formula, and use the marginals.) Are and independent?

Question 02 - Joint PDF

Suppose and have the following joint PDF: (a) Compute . (b) Compute .

(c) Set up integral(s) for the CDF of . (d) Set up integral(s) for the CDF of .

Question 03 - Sum of continuous variables

(a) State the sum rule for variance:

(b) Let and . Assume and are independent. Note: The PDF of is for and for .

Calculate the PDF of by performing a convolution.

Question 04 - Iterated Expectation

(a) State the Iterated Expectation Theorem (using any variables you like):

Now: Let . Recall the data: and . Let be uniformly distributed on the interval . (Notice that depends on .)

(b) Find . (c) Find . Hint for both: Step 1: find .