Name:
- A day trader buys an option on a stock that will return $100 profit
if the stock goes up today and lose $400 if it goes down. If the trader
thinks there is a 75% chance that the stock will go up,
- What is the expected value of the option’s profit?
\(E(x) = -25\)
- What is the standard deviation of the option’s profit?
\(SD = 227.42\)
- What do you think of this option?
With a negative expected value and a large standard deviation this
looks like a poor investment.
|
Profit
|
Prob.
|
Profit*Prob
|
Profit - E(x)
|
(Profit-E(x))^2
|
Prob(Profit-E(x))^2
|
|
100
|
0.75
|
75
|
125
|
15625
|
11718.75
|
|
-400
|
0.25
|
-100
|
-400
|
160000
|
40000
|
|
|
E(x)
|
-25
|
|
Var(x)
|
51718.75
|
|
|
|
|
|
SD(x)
|
227.42
|
- A survey reported that fifty percent of Americans believed the
country was in a recession. For a sample of 20 Americans, make the
following calculations.
- Compute the probability that exactly 12 people believed the country
was in a recession.
\(P(x = 12) = 0.12\)
- Compute the probability that no more than 5 people believed the
country was in a recession.
\(P(x \leq 5) = 0.021\)
- How many people would you expect to say the country was in a
recession?
\(E(x) = 10\)
- Compute the standard deviation of the number of people who believed
the country was in a recession.
\(SD(x) = 2.236\)
- A recent study by the department of education concluded that the
student loan default rate is 11%. Suppose a random sample of 50 students
are selected. Use this information to answer the following
questions.
- What is the expected value and standard deviation for the number of
students out of 50 that would default on their loans?
\(E(x) = 5.5\) \(SD(x) = 2.212\)
- What is the probability that exactly 5 out of the 50 students
default on their loans?
\(P(x=5) = 0.18\)
- What is the probability that less than 5 out of the 50 students
default on their student loans?
\(P(x<5) = 0.344\)
- What is the probability that more than 10 out of the 50 students
default on their student loans?
\(P(x>10) = 0.018\)
- Assume the percentage change in monthly sales at a company are
distributed uniformly with a = -5% and b = 8%.
- What is the probability that the change in sales will be less than
2%?
\(P(x < 2\%) = 0.538\)
- What is the probability that the change in sales will be greater
than 4%?
\(P(x > 4\%) = 0.308\)
- What is the probability that the change in sales will be between -2%
and 2%?
\(P(-2\% \leq x \leq 2\%) =
0.308\)
- Suppose sales at a car dealership are distributed normally by month
with a mean of $98,000 and a standard deviation of $14,000. Answer the
following questions.
- What is the probability that sales will exceed $112,000?
\(P(x \geq 112000) = 0.159\)
- What is the probability that sales will be less than $87,000?
\(P(x \leq 87000) = 0.216\)
- What is the probability that sales will be within plus and minus one
standard deviation of the mean?
\(P(84000 \leq x \leq 112000) =
0.683\)
- Given that z is a standard normal random variable, compute the
following probabilities.
- \(P(−2 \leq z \leq 2) =
0.954\)
- \(P(0.5 \leq z \leq 1.2) =
0.193\)
- \(P(−1.75 \leq z \leq −1.25) =
0.066\)
- \(P(z \leq −1.0) = 0.159\)
- \(P(z \geq −1.0) = 0.841\)
- \(P(z \leq 1.96) = 0.975\)
- The average return for companies making up the S&P 500 is 8%,
and the standard deviation is 12%. Assume stock returns are normally
distributed.
- What is the probability a company will have a stock return of at
least 6%?
\(P(x \geq 0.06) = 0.566\)
- What is the probability a company will have a stock price no higher
than 2%?
\(P(x \leq 0.02) = 0.309\)
- What stock return would put a company in the top 10% of
returns?
\(P(x \leq r) = 0.90\), \(r = 0.234\)