Download Math 115a – Exam 2 – Section 2 – Fall 00

Math 115a –Solutions to Sample Exam 2 – Section 25 – Spring 01
1. (14 points) $500 is deposited in an account that pays 6% interest.
(a) How much money is in the account at the end of 4 years, if interest is compounded monthly?
.06 4 12
)
 635.24
12
(b) What is the effective annual yield?
0.06 12
y  (1 
)  1  0.06168  6.168%
12
2. (12 points) Let H be the height of a student randomly selected from the U of A student body. Five
students are chosen at random and their heights in inches are found to be 60, 52, 62, 53 and 57.
500(1 
(a) Use this sample to estimate the mean of H, E(H).
(60  52  62  53  57) / 5  56.8
(b) Do you think your estimate is extremely accurate? Explain why or why not.
No, it is not extremely accurate. The sample is far too small to give an accurate estimate. A much larger
sample would need to be used to get an accurate estimate.
3. (12 points)
(a) If interest is compounded continuously at 7%, what is the present value of a payment of $10,000 ten
years from now?
10,000e 0.07 10  4965.85
(b) If a 30 week option is worth $5 on its expiration date, what is it worth at the start of the 30 weeks if the risk
free interest rate is 5%?
We assume the interest is compounded continuously.
5e 0.05 30 / 52  4.86
4. (20 points) For each graph, select the statement that best describes the function.
a.
b.
c.
d.
e.
The function is a probability mass function for a finite random variable.
The function is a probability density function for a continuous random variable.
The function is a cumulative distribution function for a finite random variable.
The function is a cumulative distribution function for a continuous random variable.
None of the above.
B
D
E
C
5. (15 points) The graph of the c.d.f. of a finite random variable X is shown at the bottom of the page.
(a) Find its possible values and the probability of each of these values.
The possible values are where the cdf jumps. So they are 1,3,4, and 5. Their probabilities are the heights
of the jumps. So the the pmf is
x
P(X = x)
1
0.2
3
0.3
4
0.1
5
0.4
(b) Find the mean or expected value of X.
The mean is 1 0.2  3  0.3  4  0.1  5  0.4 =3.5
6. (15 points) The length of a call to a software company’s help line is modeled by a continuous random
variable that is uniformly distributed between 0 mins and 5 mins.
(a) Draw graphs of the p.d.f. and c.d.f. for this random variable. Be sure to label your axes.
1/5
0
5
pdf
1
0
5
cdf
(b) Find the probability a call lasts between 2 and 4 mins.
1
2
( 4  2) 
5
5
7. (12 points) We have a 25 week call option on Ford stock. The strike price is $28. When the option expires
the stock price will be $25 with probability 20%, $28 with probability 25%, $30 with probability 30% and $32
with probability 25%. What is the expected value of the option on the expiration date?
0  0.2  0  0.25  2  0.3  4  0.25  1.60