Download Practice Set 2: Binomial and normal random variables No due date

Practice Set 2:
Binomial and normal random variables
No due date; personal edification only.
1. Answer the following statements TRUE or FALSE, providing a succinct explanation of your
reasoning.
(a) A binomial distribution with n = 100 and p = 0.65 is best approximated by a normal
distribution with mean µ = 65 and variance σ 2 = 18.
(b) The probability that a standard normal random variable is more than 1.96 standard
deviations from the mean is 0.05.
(c) If X ∼ Bin(N, p), then V(X) = p(1 − p)/N .
(d) The correlation of two random variables is always between -1 and 1.
(e) If X ∼ Bin(7, 0.1), Y ∼ Bin(3, 0.1), and X ⊥
⊥ Y , then X + Y ∼ Bin(10, 0.1).
(f) If X ∼ Bin(5, 0.1), Y ∼ Bin(2, 0.9), and X ⊥
⊥ Y , then X + Y ∼ Bin(7, 0.5).
(g) If X ∼ Bin(5, 0.1), Y ∼ Bin(2, 0.9), and X ⊥
⊥ Y , then E(X + Y ) = 2.3 and V(X +
Y ) = 0.63.
(h) If X is normally distributed with mean 2 and standard deviation 3, then P(−2.5 ≤ X ≤
6.5) is a number between 95% and 99.7%.
(i) Assuming independent at-bats, a player hitting 0.344 has probability of about 12% of
getting more than 2 hits in 4 at bats.
(j) Suppose X and Y are independent random variables and V(X) = 6 and V(Y ) = 6.
Then V(X + Y ) = V(2X).
(k) Suppose that two investments A and B have the same mean, but different standard
deviations. Then a diversified portfolio that includes both A and B will have a higher
expected return and lower variance than either investment alone.
(l) Historically, 15% of chips manufactured by a computer company are defective. The
probability of a random sample of 10 chips containing exactly one defect is 0.15.
(m) Let investment X have mean return 5% and a standard deviation of 5% and investment
Y have a mean return of 10% with a standard deviation of 6%. Suppose that the correlation between returns is zero. Then it is possible to find a portfolio with higher mean
and lower variance than X.
(n) A normal distribution with mean 4 and standard deviation 3.6 will provide a good approximation to a binomial random variable with parameters N = 40 and p = 0.1.
2. Consider three stocks A, B and C. Assume that each stock has normally distributed returns,
denoted by the random variables XA , XB and XC which have the following means and
standard deviations:
µA = 2%,σA = 1%
µB = 5%,σB = 10%
µC = 20%,σC = 20%
Assume also that cor(XA , XB ) = 0, cor(XA , XC ) = 0 but cor(XB , XC ) = 0.8. For example,
B and C might both be tech stocks, while A is a canned fruit concern.
Now, consider the following three two-stock portfolios, Y1 , Y2 and Y3 made up of various
combination of stocks A, B and C.
Y1 = 0.3XA + 0.7XB
Y2 = 0.6XA + 0.4XC
Y3 = 0.5XC + 0.5XB .
(a) Find the expected values of Y1 , Y2 and Y3 .
(b) Find the variances of Y1 , Y2 and Y3 .
(c) Give ranges for the returns of each portfolio Y1 , Y2 and Y3 which will contain approximately 68% of the observed returns over time, assuming the above distributions.
3. Consider a professional baseball player with a drinking problem. His probability of getting
a hit on any given at bat depends on which of three mutually exclusive physical states he
happens to be in: drunk, sober, or hung-over. Assume that he must be in one of these states.
When he’s drunk he only gets a hit 5% of the time. When he’s sober he gets a hit 30% of the
time and when he’s hung-over he gets a hit 21% of the time. Over the course of a season he
plays drunk 15% of the time and hung-over 5% of the time.
(a) What is his overall probability of getting a hit?
(b) What is the probability that he is hung-over, given that he got a hit?
(c) In a game where he has 3 at-bats, what is his probability of getting exactly 2 hits (assuming the probability of a hit in each at bat is independent of the other at bats)?
(d) Given that in a game where he had 3 at-bats he got exactly 2 hits, what is the probability
that he was sober? (Hint: you computed the necessary denominator for this calculation
in part c above.)
4. Consider the following game. You pick two distinct numbers between 1 to 6 (inclusive). You
roll three dice. If neither of your numbers appears on any dice, you lose $1. If exactly one die
shows either of your numbers, you win $1. If exactly two dice show either of your numbers,
you win $2. And if all three dice show one of your numbers, you win $3.
So your profit or loss on each roll is either −1, 1, 2 or 3.
(a) Fill in the blanks in the table.
X
-1
1
2
3
Pr(X)
(b) Compute the expected value of the game, E(X).