代写ECON 503, FALL 2022 SAMPLE MIDTERM EXAM代做迭代

ECON 503, FALL 2022

SAMPLE MIDTERM EXAM

Problem 1

(Parts a. and b. are past midterm questions. I have deliberately left the comments on the common mistakes as I thought you might find them useful)

a. Show that for two events A and C the following holds: if A C (A is a subset of C),

then

P(A) ≤ P(C).

(8 points)

b. Use the result from part a. and Bayes’ rule to show that if P(A)  >  0 then

P[(A ∩ B)  l A] ≥ P[(A ∩ B)  l (A U B)].

(12 points)

In order to get full points you need to provide a formal proof supporting your answers.

c. You are going to play two games of chess with an opponent whom you have never played against. Your opponent is equally likely to be a beginner, an intermediate, or a master; depending on this your chances of winning an individual game are 0. 9, 0. 5 or 0. 3, respectively.

c1. What is the probability to win the first game? (5 points)

c2. Congratulations: you won the first game! Given this, what is the probability that you will win the second game?

Assume that conditional on the skill level of your opponent the outcomes of each game are independent. (15 points)

Problem 2

(This is a past midterm exam question. The hint was provided.)           (15 points in total)

Suppose there are two urns containing balls; call these urns A and B. The balls in urn A are labelled with either the number 1 or the the number 2, with an equal share of each type. The balls in urn B are labelled with either the number 2, or 3, or 4, or 5, with an equal share of each type of balls.

Consider a random variable X obtained by the the following experiement. Flip a coin. If the coin comes up Heads, then draw a ball from urn A and define X as the number on that ball. If If the coin comes up Tails, then draw a ball from urn B and let X be the number on  that ball.

a. Find the PWF ofX.        (10 points)

Hint: Since the balls numbered 1 and 2 in urn A are of equal share each, then they are equally likely; similarly, for balls numbered 2, or 3, or 4, or 5 in urn B.

b. Find the CDF ofX, FX(x) and draw its graph. (5 points)

Problem 3

(20 points in total)

A continuous random variable X has a PDF given by:

 

a. Show that c = 2/3 makes fX(x) a valid PDF, and impose this value in all subsequent parts of the question. (3 points)

b. Find the CDF of X, FX(x). (5 points)

c. Find P(1 ≤x ≤2). (3 points)

d. Find P(x < 2/1  lx <1). No need to calculate it, just give a numeric expression (4 points)

e. Find the expected value (mean) of X. (5 points)

Problem 4

(20 points)

A continuous random variable X has a PDF given by:

Find the PDF of the transformation

Problem 5

(10 points)

A continuous random variable X has a PDF given by:

Find E(X) as a function of the parameter a. What happens to E(X) as a  →  +∞?

Problem 6 (Problem 3.3.1 from the textbook)

(10 points each; 20 points in total)

An urn contains five balls numbered 1 to 5. Two balls are drawn simultaneously.

a1. Let X be the larger (maximum) of the two numbers drawn. Find the PMF of X.

a2. Let Y be the sum of the two numbers drawn. Find the PMF of Y.