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game show host shows you three doors D1, D2, D3, tells you that behind one door is a car and behind the other two are goats

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Homework 3:

• Write your full name, “Homework 3”, and the date at the top of the first page.

• Show all work, including each step of your solution, to earn maximal partial credit.

• Each question has multiple parts. Write legibly and neatly. Box your final answers.

• Use Genius Scan or a similar application to convert your solutions to .pdf format.

• Submit a single .pdf file to Gradescope under the assignment “Homework 3”.

• If you have any questions, email me or come to office hours (WF 11:00am-12:00pm)

• You are encouraged to work together (on Piazza) but must write up your own solutions.

Assignment (4 Problems: 20+ 20 + 20 + 40 = 100 points total.)

Q Problem 1 Recall that the cumulative distribution function of any random variable X

on a probability space (S, P ) is the function FX : R → [0, 1] defined by FX(b) = P (X ≤ b).

• 1.1 [5 points] If X is continuous with probability density function fX(x), show that

f (b) = lim FX(b + h) − FX(b) .

h→0 h

• 1.2 [5 points] Let X be uniformly distributed on (0, 1) as in §2.3.1 [Ross]. Find FX(x).

• 1.3 [10 points] Let X1, X2, X3 be independent and each uniformly distributed over the interval (0, 1). Find FY (y) for Y = max(X1, X2, X3).

Q Problem 2 Let X be an exponential random variable with λ = 2 as in §2.3.2 [Ross].

• 2.1 [10 points] Find P (X > 9).

• 2.2 [10 points] Find P (X > 10|X > 1).

Q Problem 3 In Lecture 6, for the triangle Ω with vertices (0, 0), (2, 0), (0, 2) we saw that X and Y defined by the sample outcome →v = (X, Y ) in Ω are dependent random variables according to Punif. X, Y are jointly continuous with joint probability density function fX,Y

(x, y) = ( 1if (X, Y ) ∈ Ω

0 if (X, Y ) /∈ Ω.

• 3.1 [10 points] Calculate P (X ≥ 1|Y ≤ 1).

• 3.2 [10 points] Calculate Cov[X, Y ].

Q Problem 4 Xander flips a fair coin C1 and records the outcome as:

= 1 if C1 = H

0 if C1 = T

Consider the two scenarios:

(Scenario 1) Yolanda flips a fair coin C2 independently of C1 and writes down:

= 1 if C2 = H

0 if C2 = T

(Scenario 2) Yolanda does not flip C2 but instead copies the outcome of Xander’s flip of C1:

= 1 if C1 = H

0 if C1 = T

Recall the joint probability mass function of discrete X, Y is

pX,Y (x, y) = P (X = x and Y = y)

and may be represented as a table so pX(x), pY (y) appear as marginal distributions.

• 4.1 [5 points] Find pX,Y (1, 0) in Scenario 1.

• 4.2 [5 points] Find pX,Y (1, 0) in Scenario 2.

• 4.3 [5 points] Find pX,Y (x, y) in Scenario 1. Write your answer as a table.

• 4.4 [5 points] Find pX,Y (x, y) in Scenario 2. Write your answer as a table.

• 4.5 [5 points] Find Cov[X, Y ] in Scenario 1.

• 4.6 [5 points] Find Cov[X, Y ] in Scenario 2.

• 4.7 [5 points] Find pY (y) in Scenario 1.

• 4.8 [5 points] Find pY (y) in Scenario 2.

Q Bonus [X points] Use conditional probabilities to solve the Monty Hall Problem:

A game show host shows you three doors D1, D2, D3, tells you that behind one door is a car and behind the other two are goats, and asks you to guess which door the car is behind. You pick D1. The host, who knows where the car is, opens up door D3 and reveals a goat behind D3. The host then gives you the option of changing your guess to D2. Is it to your advantage to change your guess to D2 or to stick with your original guess D1?

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