ECN 422
PS 2
Due in class on Oct 3rd

Part 1 Summary

Read textbook Chapter 3 and write a minimum one-page summary

Part 2 Questions

Expected Value and the Variance

(Question 1) I toss a fair coin, twice.  I pay you one dollar for every heads and nothing for every tails.  Denote by X the random variable associated with your payo§ playing this game.

a) What is the domain of X?

b) Find E[X].

c) Find Var[X].

(Question 2) I roll a fair die. You pay me one dollar if the outcome is 1 and I pay you 20 cents if the outcome is 2 or larger.  Denote by X the random variable associated with your payo§ playing this game.

a) What is the domain of X?

b) Is this game fair? Find E[X].

c) Find Var[X].

d) Now, suppose that, instead, you pay me 10 dollars if the outcome is 1 and I pay you 2 dollars if the outcome is 2 or larger.1    Denote by Y  the random variable associated with your payo§ playing this game.  Find E[Y] and Var[Y].

(Question 3) You roll a pair of fair dice.  Let Z1  be the random variable associated with the outcome of the Örst die and Z2  be the random variable associated with the outcome of the second die.

a) Suppose that both dice are six-faced. What is the chance that the largest outcome will be 6?

b) What is the chance that the largest outcome will be 5?

c) Now, suppose that both dice are ten-faced. What is the chance that the largest outcome will be 10?

d) What is the chance that the largest outcome will be 9?

(Question 4) In a casino, there is a game that goes as follows:  They roll 2 six-faced dice.  If the largest of the outcomes is 5 or 6, they pay you one dollar. If the largest of the outcomes is 4 or less, you pay them 1.25 dollars.

a) Is this game fair?  Are you going to make/lose money in the long-run? Find E[Y] where Y is the random variable associated with your payo§.2

b) How risky is this game? Find Var(Y).

(Question 5) Consider a disease whose presence can be identiÖed by carrying out a blood test. Assume that the probability that a randomly selected indi- vidual has the disease is 10%.  Suppose that 3 individuals are independently selected for testing.  One way to proceed is to carry out a separate test on each of the 3 individuals in the sample.

A potentially more economical approach, group testing, was introduced dur- ing World War II to identify syphilitic men among army inductees.  First, take a part of each blood sample, combine these specimens, and carry out a single test. If no one has the disease, the result will be negative, and only the one test is required.  If at least one individual is sick, the test on the combined sample will yield a positive result, in which case the 3 individual tests are then carried out.

Is this procedure better on average than simply testing everyone?

Conditional Probabilities and the BayesíRule

(Question 6) Suppose an individual is randomly selected from the popula- tion of all adult males living in the United States.  Let A be the event that the selected individual is over 6 ft in height, and let B be the event that the selected individual is a professional basketball player. Which do you to think is larger, P (AjB) or P (BjA)? Why?

(Question 7) Alice has two coins in her pocket, a fair coin (head on one side and tail on the other side) and a two-headed coin.

a) She randomly picks one of the coins from her pocket, tosses it, and obtains tails.  What is the probability that she áipped the fair coin?  Explain your answer.

b) She randomly picks one of the coins from her pocket, tosses it, and ob- tains head. What is the probability that she áipped the fair coin?

c) She randomly picks one of the coins from her pocket, tosses it twice, and obtains two heads. What is the probability that she áipped the fair coin?   d) She randomly picks one of the coins from her pocket, tosses it 5 times, and obtains Öve heads.  What is the probability that she áipped the fair coin?

e) Compare your answer from parts (b) and (d).  Can we say for sure that she is tossing the two-headed coin, after a string of 5 heads?  Are we at least more conÖdent in that event than we were after a single toss?

Useful formulas:

Conditional Probability:  P [AjB] = P [A \ B]=P [B]

BayesíRule:  P [AjB] = P [BjA]:P [A]=P [B]

Law of total probability:  P [B] = P [BjA]:P [A] + P [BjA0]:P [A0]

Expected Value:  E[X] =PP [X = xi]xi

Expected Value of functions of X:  E[h(X)] =PP [X = xi]h(xi)

Variance:  Var[X] = E[(X - E[X])]2

Variance (shortcut formula):  Var[X] = E[X2] - E[X]2