# HOMEWORK 4 – STAT 3360 – FALL 2015

Question

HOMEWORK 4 – STAT 3360 – FALL 2015

DUE BY SEPTEMBER 29

PLEASE, SUBMIT THIS DOCUMENT AND SHOW YOUR FIRST AND LAST NAME BELOW!

STAPLE OR PAPERCLIP YOUR WORK, PLEASE!

1

PLEASE, READ SECTIONS 7.1 – 7.3 BEFORE SOLVING THESE EXERCISES.

SECTION [CIRCLE ONE! =>] 001 002 501

FIRST NAME LAST NAME

PROBLEM 1 [30 POINTS = 5 + 10 + 15]

THE NUMBER (X) OF CARS SOLD BY A RANDOMLY SELECTED DEALERSHIP IN THE

METROPLEX AREA DURING ONE DAY HAS A DISTRIBUTION SHOWN BELOW.

X = NUMBER OF CARS SOLD PROBABILITY

2 0.15

3 0.30

4 0.25

5 0.20

6 0.10

1. DETERMINE THE EXPECTED VALUE OF X.

E [X] =

2. WHAT IS THE VARIANCE OF X?

VAR [X] =

3. THE COMMISSION IS DEFINED (IN $1,000) AS Y = 16 + 5 X.

FIND THE EXPECTATION AND VARIANCE OF Y.

E [Y] =

VAR [Y] =

2

PROBLEM 2 [20 POINTS = 5 + 5 + 5 + 5]

TWO RANDOM VARIABLES, X AND Y, ARE INDEPENDENT. THEIR MARGINAL

DISTRIBUTIONS ARE SPECIFIED AS FOLLOWS.

X – 10 0 5

PROBABILITY 10% 40% 50%

Y – 4 6

PROBABILITY 70% 30%

1. FIND EXPECTATION FOR EACH VARIABLE.

E [X] =

E [Y] =

2. FIND THE VARIANCE FOR EACH VARIABLE.

VAR [X] =

VAR [Y] =

3. WHAT IS THE EXPECTATION OF THE NEW RANDOM VARIABLE, Z = 3X – 2Y?

E [Z] =

4. FIND THE VARIANCE OF Z (= 3X – 2Y).

VAR [Z] =

3

PROBLEM 3 [50 POINTS = 10+15+15+10]

SUZY IS A STOCK MARKET ANALYST. SHE WANTS TO BUILD A PORTFOLIO INVESTING

50% INTO ONE STOCK AND 50% INTO THE OTHER. THREE STOCKS (I, II, AND III) ARE

CONSIDERED FOR SELECTION. THEIR EXPECTED ROI VALUES (IN CENTS PER DOLLAR)

AND CORRESPONDING STANDARD DEVIATIONS (IN CENTS PER DOLLAR) ARE LISTED IN

THE TABLE BELOW.

STOCK ROI EXPECTED ROI STANDARD DEVIATION

I X 14 5

II Y 14 5

III Z 14 5

IN ADDITION, SUZY ASSUMES THAT THE CORRELATION BETWEEN THE ROI FOR STOCK I

AND THAT FOR STOCK II IS C [X, Y] = – 0.8, WHILE THE ROI FOR STOCK III IS

UNCORRELATED WITH THAT FOR STOCK I (THAT IS C [Z, X] = 0). CORRELATION

BETWEEN THE ROI FOR STOCK II AND THAT FOR STOCK III WAS SET AS C [Y, Z] = 0.6.

SHE WANTS TO MAKE A CHOICE BETWEEN TWO TYPES OF SCENARIO.

FIRST SCENARIO OPERATES WITH STOCKS I AND II, SO THE ROI FOR THE PORTFOLIO

WILL BE FOUND AS T = 0.5 X + 0.5 Y.

THE ALTERNATIVE SCENARIO WILL OPERATE WITH STOCKS II AND III, SO THE ROI IN

THIS CASE WILL BE W = 0.5 Y + 0.5 Z.

HELP SUZY COMPARE TWO CHOICES FOR THE PORTFOLIO BY ANSWERING THE

FOLLOWING QUESTIONS.

1. WHAT IS THE EXPECTED ROI FOR T AND THAT FOR W?

E [T] =

E [W] =

2. WHAT IS THE VOLATILITY (STANDARD DEVIATION) ASSOCIATED WITH THE

PORTFOLIO CHARACTERIZED BY A RANDOM VARIABLE T?

SD [T] =

3. WHAT IS THE VOLATILITY (STANDARD DEVIATION) ASSOCIATED WITH THE

PORTFOLIO CHARACTERIZED BY A RANDOM VARIABLE W?

SD [W] =

4. BASED ON VOLATILITY COMPARISON, WHICH SCENARIO (T OR W) WOULD YOU

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