STAT 334 ASSIGNMENT 1

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STAT 334

ASSIGNMENT 1

This assignment has 12 questions, and the total points you can score is 101.

The answers should be uploaded on Crowdmark. Check for the link on Learn or in your email.

All relevant work needs to be shown. Correct answers without the proper explanation will get zero credit.

You can consult your notes but not each other. You can leave your final answer in

Due Date: June 21st, Friday, 11: 59 p.m. (on Crowdmark)

1) (5) A student is taking a multiple-choice exam. There are four choices for each question. A student has studied enough so that

(i) They would know the correct answer with probability 0.25

(ii) They will be able to eliminate one incorrect answer with probability 0.25.

Otherwise, all four choices seem equally possible.

If they know the answer, they get the question right. If they don’t, they have to guess from the 3 or 4 choices.

You are the instructor for this course. If a student answers a question correctly, what is the probability that they knew the answer?

2) (3+4) (a) 4 men and 4 women are ranked according to their scores on an exam. Assume that no twoscores are the same and all possible rankings are equally likely. Let the random variable X be the highest ranking achieved by a man. What is the probability mass function of X?

(b) In a lottery a four-digit number is chosen at random from the range 0000 − 9999. A lottery ticket costs $3. You win $50 if your ticketmatches the last two digits but not the last three, $500 if your ticket matches the last three digits but not all four, and $5,000 if your ticket matches all four digits. What is the expected payoff on a lottery ticket? What is the house edge of the lottery?

(House Edge is the proportion of the price of the lottery ticket the issuer gets to pocket as (expected) profits. For example, if the expected payoff to the customer is $2, then the house edge is (3-2)/3 =33/3%)

3. (5) A card player is dealt a 13-card hand at random from a well shuffled standard deck of cards.

A hand is said to be “void in a suit” if it does not have any cards from that suit (There are four suits—diamonds, hearts, spades, and clubs. Each suit has 13 cards).

What is the probability that the hand is void in at least one suit?

4. (3+3+3) There are 6 people. Lets call them 1,2,3,4,5 and 6. Each of them has a hat. They throw their hats in a box, and then each removes one hat from it (each ordering equally likely).

Let X = number of people who get their own hat.

A pair of people is called a swapped pair if each has the other person’s hat. (There can be at most three swapped pairs)

Let Y = number of swapped pairs.

Compute the following.

a) E(X)

b) E(Y)

c) Var(X)

5. (2*4=8) Every day, I get important phone calls on my phone as a Poisson Process with parameter λG = 0.5/hour. I get spam phone calls (independently) as a Poisson Process with parameter λB = 1.5/hour.

a) Let T = amount of time until I get my first phone call (important or spam). Find the pdf of T.

b) Find the probability that I get exactly 2 spam calls during my first half hour.

c) Let S = amount of time elapsed before the fourth important phone call. Compute Var(S).

d) What is the probability that I don’t get any phone calls during my first three hours?

6. (2+2+3+3+2=12) Suppose the random variable X has the pdf

f(x) = cx, 1 < y < 5

     = 0, otherwise

a) Find the value of c.

b) Find the CDF FX(x).

c) Compute E(X) and Var(X).

d) Derive the MGF of X.

e) Find the probability that X lies between 2 and 4.

7. (4+2) Surya wants to sell his car and retire to the Himalayas. He decides to sell it to the first person who offers him at least $15,000. Assume that the offers are Exponential with mean $10,000, independent. (You can assume memorylessness, if required)

a) Find the expected number of offers Surya will receive.

b) Find the expected amount of money Surya will get for his car.

8. (2+2+4=8) Let X1, X2, X3 be independent random variables U[0,1]. Let Y = X1 + X2, and Z = X2 + X3.

a) Compute E(X1X2X3).

b) Compute Cov(Y, Z) and the correlation coefficient between Y and Z.

c) Find the pdf of Y.

9. (18) (Drawing figures may be useful here). Let (X, Y) have a joint pdf f(x, y) = cx, 0 < x < 1, 0 < y < 1, x + y < 1.

a) Find the constant c

b) FX, Y (2/1, 2/1)., where F is the CDF.

c) FX (4/3) − FX (4/1)

d) The marginal pdf of Y

e) The CDF of X.

f) The CDF (and pdf) of Z=X+Y.

10. (5+1) Suppose we have two independent Unif(0,1) distributions X, Y. Let S = XY, T = X/Y.

a) Find the joint distribution of S and T.

b) Are S and T independent?

11. (3+3+2=8) Let X be a random variable whose pdf is given by fx(x) = e-2x + 2/1e-x > 0, 0 otherwise.

a) Find the MGF of X.

b) Using (a), find the mean and variance of X.

c) If Y=4+6X, what is the MGF of Y?

12. (3+3+3=9) Suppose that X1, X2, … Xn, are independent Exponential random variables each with pdf f(x) = λ exp{−λx}, x > 0. Find the pdf of

a) Y= max {X1, X2, … Xn,}

b) Z= min {X1, X2, … Xn,}

c) W=X(2),the second order statistics of X1, X2, … Xn,.