STAT 3360 – PRACTICE EXAM 2 | Complete Solution

STAT 3360 – PRACTICE EXAM 2

THIS IS NOT THE REAL EXAM. IT CONTAINS QUESTIONS SIMILAR TO THE ONES ON THE SECOND EXAM AND IS AIMED AT HELPING YOU ENHANCE YOUR PREPARATIONS. THE REAL EXAM WILL HAVE OTHER PROBLEMS, SIMILAR BUT NOT NECESSARILY THE SAME! PLEASE, SOLVE THE PROBLEMS ON YOUR OWN OR ASK QUESTIONS IN CLASS. HINTS AND SOLUTIONS WILL BE PROVIDED IN CLASS PRIOR TO THE EXAM DAY.
1. [SECTION 7.5 + TABLE 2]
A CAR MARKET ANALYST ASSUMES THAT THE AVERAGE CAR SALES VOLUME FOLLOWS THE POISSON DISTRIBUTION WITH THE INTENSITY OF 0.15 OF A CAR PER DAY. IF X DENOTES HOW MANY CARS WILL BE SOLD BY A RANDOMLY SELECTED CAR DEALER FOR A PERIOD OF 30 DAYS, FIND THE FOLLOWING PROBABILITIES.
A. WHAT IS THE CHANCE TO SELL AT LEAST 5 CARS? P [X ≥ 5] =
B. WHAT IS THE CHANCE TO SELL FEWER THAN 8 CARS? P [X < 8] =
C. WHAT IS THE CHANCE TO SELL AT MOST 8 AND AT LEAST 3 CARS? P [3 ≤ X ≤ 8] =
D. WHAT IS THE CHANCE TO SELL MORE THAN 5 CARS? P [X > 5] =
2. [SECTION 7.4]
TOM AND JERRY PLAY A SET OF 6 TENNIS GAMES. ASSUME THAT IN A SINGLE GAME TOM HAS A 40% CHANCE TO WIN (AND THEREFORE, A 60% CHANCE TO LOSE). ALSO ASSUME THAT GAME RESULTS ARE INDEPENDENT. CONSIDER A RANDOM VARIABLE (Y) EQUAL TO THE NUMBER OF VICTORIES FOR TOM.
A. WHAT IS A PROBABILITY THAT TOM WINS AT MOST ONE GAME?
B. WHAT IS A PROBABILITY THAT TOM WINS AT LEAST ONE GAME?
C. FIND THE EXPECTATION, VARIANCE, AND STANDARD DEVIATION OF Y.
D. NOW IMAGINE THAT FOR EVERY WIN TOM GETS PAID $4 AND PAYS $6 FOR EVERY LOST GAME. CONSIDER A VARIABLE (W) THAT MEASURES TOM’S BALANCE. FIND THE EXPECTATION AND VARIANCE OF W.
3. [SECTION 7.4]
TOM AND JERRY PLAY A SET OF 5 TENNIS GAMES. ASSUME THAT IN A SINGLE GAME TOM HAS A 70% CHANCE TO WIN (AND THEREFORE, A 30% CHANCE TO LOSE). ALSO ASSUME THAT GAME RESULTS ARE INDEPENDENT. CONSIDER A RANDOM VARIABLE (Y) EQUAL TO THE NUMBER OF VICTORIES FOR TOM.
A. WHAT IS A PROBABILITY THAT TOM WINS AT MOST ONE GAME?
B. WHAT IS A PROBABILITY THAT TOM WINS AT LEAST ONE GAME?
C. FIND THE EXPECTATION, VARIANCE, AND STANDARD DEVIATION OF Y.
D. NOW IMAGINE THAT FOR EVERY WIN TOM GETS PAID $5 AND PAYS $4 FOR EVERY LOST GAME. CONSIDER A VARIABLE (W) THAT MEASURES TOM’S BALANCE. FIND THE EXPECTATION AND VARIANCE OF W. 1


4. [SECTIONS 7.1 – 7.3]
SUZY, A STOCK MARKET ANALYST, WANTS TO BUILD A PORTFOLIO INVESTING 40% INTO ONE STOCK AND 60% INTO THE OTHER. TWO STOCKS (I AND II) 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
12
5
II
Y
12
5
IN ADDITION, SUZY ASSUMES THAT THE CORRELATION BETWEEN THE ROI FOR STOCK I AND THAT FOR STOCK II IS ρ [X, Y] = – 0.8. SUZY FOUND THE ROI FOR THE PORTFOLIO AS
T = 0.4 X + 0.6 Y.
ASSESS THE PORTFOLIO BY FINDING ITS EXPECTED ROI AND VOLATILITY.
E [T] =
SD [T] =
5. [SECTIONS 7.1 – 7.3]
SAM, A STOCK MARKET ANALYST, WANTS TO BUILD A PORTFOLIO INVESTING 60% INTO ONE STOCK AND 40% INTO THE OTHER. TWO STOCKS (I AND II) 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
10
4
II
Y
10
4
IN ADDITION, SAM ASSUMES THAT THE CORRELATION BETWEEN THE ROI FOR STOCK I AND THAT FOR STOCK II IS ρ [X, Y] = – 0.25. SAM FOUND THE ROI FOR THE PORTFOLIO AS
W = 0.6 X + 0.4 Y.
ASSESS THE PORTFOLIO BY FINDING ITS EXPECTED ROI AND VOLATILITY.
E [W] =
SD [W] =


6. [SECTION 7.5 + TABLE 2]
TINA, A SMALL BARBER SHOP OWNER, WANTS TO ASSESS RISKS OF LOSING PROSPECTIVE CUSTOMERS. SHE HAS TWO HAIR DRESSERS AND THREE WAITING SEATS. THEREFORE, IT IS IMPORTANT TO EVALUATE THE POSSIBILITY TO HAVE AN EXCESSIVE NUMBER OF CUSTOMERS.
TINA INVITED A BUSINESS ANALYST TO HELP HER ASSESS RISKS. TRENT (THE ANALYST) ASSUMED THAT THE CUSTOMERS ARRIVE AT THE SHOP WITH THE EXPECTED INTENSITY OF 1 CUSTOMER PER 15 MINUTES. HE ASSUMES THAT Y IS A COUNT OF CUSTOMERS WITHIN ONE HOUR.
A. WHAT IS THE PROBABILITY DISTRIBUTION OF Y? FIND ITS PARAMETER. (EXPECTED VALUE OF Y) =
B. WHAT IS THE PROBABILITY OF HAVING A REASONABLE LOAD, 2 TO 5 CUSTOMERS PER ONE HOUR? P [2 ≤ Y ≤ 5] =
C. WHAT IS THE CHANCE OF HAVING A HEAVY LOAD, MORE THAN 10 CUSTOMERS? P [Y > 10] =
D. WHAT IS THE CHANCE OF HAVING 5 OR 6 CUSTOMERS? P [5 ≤ Y ≤ 6] =
7. [SECTIONS 7.1 – 7.2]
A PIZZA DELIVERY OWNER FOUND THAT THE NUMBER (X) OF PIZZAS ORDERED BY STUDENTS RESIDING IN THE DORMITORY HAS THE DISTRIBUTION DESCRIBED BY THE TABLE BELOW. NOTICE THAT THE MAXIMAL ORDER WAS 5 PIZZAS.
X
0
1
2
3
4
5
TOTAL
PROBABILITY
0.15
0.20
0.25
0.10
0.20
1.00
A. WHAT IS THE PROBABILITY OF AN ORDER OF 5 PIZZAS? P [X = 5] =
B. FIND THE EXPECTED NUMBER OF PIZZAS. E [X] =
C. FIND THE VARIANCE OF THE NUMBER OF PIZZAS. VAR [X] =
D. IN ORDER TO REMAIN IN BUSINESS, THE OWNER WANTS TO ENSURE THAT THE EXPECTED REVENUE (Y) MEASURED AS Y = 20 X – 40, WILL REMAIN POSITIVE. EVALUATE THE EXPECTATION AND VARIANCE OF Y. E [Y] = VAR [Y] =