Math 115a
Summer 2002
Final Exam
Professor Hermi
Name: ______________________________
Please show all steps and all work to receive full credit.
1.
(a) At what rate does it take $15,000 to grow to $60,000 with interest
compounded monthly for 20 years?
(b) What would this rate be if the money was compounded continuously?
2.
Consider an experiment of three coins being tossed at once.
(a) Give the sample space in terms of heads and tails.
(b) Let X be the random variable for the number of heads that show up in one
trial. What are the possible outcomes for X?
(c) Compute P(X=2).
(d) Compute  X .
(e) Give the formula for the c.d.f. of X.
3.
Compute the following sums:
4
(a)  2  ( j 2  j  3) .
j 1


100
(b)  3.1k 1  3.1k .
k 2
4.
Consider the experiment of rolling a fair die twice. All of the 36 outcomes in the
sample space S shown below are equally likely. Consider the following events:
A is the event that the sum of the faces is greater than 7.
B is the event that the sum of the faces is even.
C is the event that the sum of the faces is less than 5.
D is the event that the first toss is an even number.
E is the event that the second toss is an odd number.
ì (1,1)
ï ( 2 ,1)
ï
ï ( 3,1)
Sí
ï ( 4 ,1)
ï (5,1)
ï
î ( 6,1)
(1, 2 )
(2, 2)
( 3, 2 )
(4, 2)
(5, 2 )
( 6, 2 )
(1, 3)
( 2 , 3)
( 3, 3)
( 4 , 3)
(5, 3)
( 6, 3)
(1, 4 )
(2, 4)
( 3, 4 )
(4, 4)
(5, 4 )
( 6, 4 )
(1, 5)
( 2 , 5)
( 3, 5)
( 4 , 5)
(5, 5)
( 6, 5)
(1, 6) ü
( 2 , 6) ï
ï
( 3, 6) ï
ý
( 4 , 6) ï
(5, 6) ï
ï
( 6, 6) þ
Find
(a) P( A  D) =_______
(b) P(B|C) =_______
(c) Find the probability that the sum of the faces is greater than 7 or the sum of the faces
is even.______
(d) Find the probability that the sum of the faces is greater than 7 or the sum of the faces
is even, but not both.__________
4.
Let A be the age of a randomly selected student in your Math 115a class. You
find that the ages of the seven students sitting in your row are 20, 21, 25, 19, 18,
21 and 20 years.
(a) Use this information to estimate E(A).
(b) Use this information to estimate FA(20).
(c) Give a practical explanation of the value FA(20) in terms of this specific
problem.
5.
Let N denote the event that the NASDAQ index goes up on a given day in 2002,
and let D denote the event that the Dow Jones Industrial Average goes up on the
same day. Based on the following Venn diagrams, shade in the areas
corresponding to the event that:
(a) NASDAQ goes up, but not the Dow Jones
(b) Neither NASDAQ nor the Dow Jones goes up.
(c) One of the two goes up but not both.
(a)
(b)
(c)
6.
Suppose that 60% of visitors to a store are women and the rest are men. Also,
20% of the women who visit make a purchase, and 30% of the men who visit
make a purchase.
(a) Draw a tree diagram that describes the events and the probabilities.
(b) Give the probability that the next customer who comes into the store is a man
who will buy something.
(c) Use Bayes’ theorem to find the probability that the next visitor who buys
something is a man.
7.
Given an annual risk-free rate of investment of 2.10%, compute the following:
(a) Weekly risk-free rate.
(b) Weekly risk-free ratio.
8.
Consider the p.d.f. shown below.
(a) Shade the area pertaining to P(3  X  5) and compute it.
(b) Compute FX (4) .
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
9.
Consider the bank records shown in the Microsoft Excel table.
(a) Fill in the blanks in the VLOOKUP command to show how you would use
that command and RANDBETWEEN to choose a record at random, and
return the number of years in business.
VLOOKUP( RANDBETWEEN(
,
),
,
)
(b) Fill in the value returned in E2 if the formula entered into that cell is the
formula
IF(D2 > C2, 25,0).
(c) If you did a “drag and drop” down to the next two cells from E1, what would
appear in the cells?
10. For each graph, select the statement that is most likely to be correct.
(i). The function is a probability mass function for a discrete random variable.
(ii). The function is a probability density function for a continuous random variable.
(iii). The function is a cumulative distribution function for a discrete random variable.
(iv). The function is a cumulative distribution function for a continuous random
variable.
(v). None of the above.
0.9
0.8
0.7
0.6
1
0.5
0.4
0.3
0.5
0.2
0.1
0
0
-1
1
2
3
4
5
0
6
(a) _____
(b) _____
1.5
1.5
1
1
0.5
0.5
0
0
-1
(c) _____
0
1
(d) _____
1
Formulas
r

P  F  1  
n

 nt
1

e  lim 1  
m
m 
r

F  P  1  
n

F = Pert
C
 x  P( X  x )
all x
P = Fe-rt
1




n
r  n   1  y   1




E(X) = np
ì0 if x  0
FX ( x)  í
x /
if x  0
î1  e
P A | B1   PB1 
P A | B1   PB1   P A | B2   PB2 
E C  F C  E  F 
E( X ) 
n
r

y  1    1
 n
m
ì0 if x  0
ï
f X ( x)  í 1  x / 
e
if 0  x
ï
î
PB1 | A 
n t
PB1 | A 
PE  F   P( E)  P( F )  PE  F 
E C  F C  E  F C
E(X) = 
P A | B1   PB1 
P A | B1   PB1   P A | B2   PB2 
 
P E C  1  P( E )
PE  F   PE) P( F 