Business Statistics 41000: Homework # 2

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Warren Moody
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1 Business Statistics 41000: Homework # 2 Drew Creal Due date: At the beginning of lecture # 5 Remarks: These questions cover Lectures #3 and #4. Question # 1. Discrete Random Variables and Their Distributions Suppose the probability distribution of the random variable X is given by the following table: x p(x) ?? We might think of X as describing the return on an asset over the next year. (a) What is P r(x =.10)? (b) Plot x versus P (x) (Feel free to do this plot by hand, you don't have to use Excel) (c) What is P r(x >.05)? (d) What is E(X)? (e) What is V ar(x)? (f ) What is the standard deviation of X? 1

2 Question # 2. Discrete Random Variables and Their Distributions Suppose you roll a standard six-sided die. Let X be the random variable which is a 1 if the die comes up 6 and 0 otherwise. (You can think of this random variable as a dice game where you need to roll a 6 to win. If you don't roll a 6, you don't really care whether you rolled a 1, 2, 3, 4, or 5, because you lost anyway.) (a) What is the distribution of the random variable X? (You can write out the probabilities, OR you can recognize that this is a `special' distribution we discussed in class...) (b) What are the mean and variance of X? (Again you can write this all out OR you can use a formula from the lecture notes to nish this problem in 10 seconds...)

3 Question # 3. Discrete Random Variables and Their Distributions Consider the following simple model of an asset return, R. Let R be the random variable with the following possible values and corresponding probabilities. r p(r) (a) What is P r(0 < R < 0.15)? (b) What is the probability that the return is NEGATIVE? (c) What is the expected return, E(R)? (d) What is σ R? (e) Plot p(r) versus r.

4 Question # 4. Marginal and Conditional Distributions of Discrete Random Variables Suppose we're looking at a given company. For simplicity assume this company is either having a good year or a bad year. We also observe whether the company meets or misses its earnings target this quarter (that is, whether or not the company's earnings per share is above or below analysts' consensus forecast). Let G = 1 if the company is having a GOOD year, and 0 otherwise. Let E = 1 if the company makes its EARNINGS target, 0 if it misses. Now suppose we've done some research on this company. Based on past history, this company has good years about 50% of the time. We also know based on past experience that in a good year, the company will make its earnings target 80% of the time, while in a bad year they make their earnings target only 40% of the time. That is, p(g = 1) =.5, p(e = 1 G = 1) =.8, and p(e = 1 G = 0) =.4 (a) Draw the tree diagram depicting the marginal distribution of G and then the conditional distributions of E G. (There is a similar graph in the lecture notes.) (b) Give the joint distribution of the G and E in the two way table format. (c) In reality, as investors we don't know at the time whether the company is having a good or bad year (the managers would always claim it's a good year). But we DO observe whether or not the company makes or misses its earnings target. If the company DOES make its earnings target, what is the probability they are actually having a good year? That is, what is P (G = 1 E = 1)?

5 Question # 5. Marginal and Conditional Distributions of Discrete Random Variables Suppose we are investors looking at two assets. Let X be the return on the rst asset and Y be the return on the second asset. For simplicity, let's say the two assets can each return 5%, 10%, or 15% in a given year. The table below shows the joint distribution of the two asset returns: X 5% 10% 15% Y 5% % % (a) What is the probability that BOTH asset returns are 10% or less? (b) What is the marginal distribution of X? (c) What is the marginal distribution of Y? (d) What is the conditional distribution of Y X = 15%? (e) What is the conditional distribution of Y X = 5%? (f ) Do you think X and Y are positively related, negatively related, or not related? (Hint: Compare your answers to parts c,d, and e.) (g) What are the mean and variance of Y? (h) What is the conditional mean E(Y X = 15%)? If you believe the rst asset will do well this year, does it aect your guess for the return on the second asset? Briey explain why this is related to your answer to part (f).

6 Question # 6. Sampling WITHOUT replacement and WITH replacment Suppose we have a group of 10 voters: 5 Democrats and 5 Republicans. We are about to pick two voters from this group of ten. Let Y 1 be 1 if the rst voter we pick is a Democrat, and 0 otherwise. Let Y 2 be 1 if the second voter we pick is a Democrat, and 0 otherwise. Suppose we sample WITHOUT replacement. This means that after we ask the rst voter whether s/he is a Democrat or Republican, we cannot choose this person again. We select the second voter at random from the nine people who are left. (NOTE: This is an important problem. Please work through it carefully.) (a) What is the probability distribution of Y 1? (Hint: It has a name!) (b) What is the conditional probability distribution of Y 2 given Y 1 = 1? (c) (d) Find the joint distribution of Y 1 and Y 2. Write in a (2 x 2) table like we did in the notes. Find the marginal probability distribution of Y 2 using your table. You should see that the marginal distributions of Y 2 and Y 1 are the same! This means that Y 2 and Y 1 are identically distributed. BUT, we can also see that the outcome of Y 1 does inuence the probabilities associated with Y 2 ; the conditional distributions are dierent from the marginal. (e) Now suppose we randomly choose a third voter (again without replacement) and let Y 3 = 1 if the third voter is a Democrat, and 0 otherwise. Find the joint distribution of (Y 1, Y 2, Y 3 ) by determining the probabilities p(y 1, y 2, y 3 ) of each possible combination of values and entering them in the table below: p(y 1, y 2, y 3 ) (0,0,0) (0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1) (1,1,0) (1,1,1)

7 Now suppose we sample WITH replacement. This means that after we ask the rst voter whether s/he is a Democrat or Republican, we ask her/him to rejoin the group, and that person may then be selected again when we randomly choose the second voter. (f ) Under these circumstances, what is the conditional probability distribution of Y 2 given Y 1 = 1? How is it related to the marginal distribution of Y 2? Give a brief explanation (2 sentences max). Remarks: Notice that Y 1 and Y 2 are independent random variables because the probabilities we assign to dierent values of Y 2 do not change based on the value of Y 1 = y 1. (In parts a-d, Y 1 and Y 2 were NOT independent.) In fact, when we sample with replacement, Y 1 and Y 2 are independent and identically distributed (abbreviated i.i.d.). We will see that i.i.d. of nice properties, and they play a very important role in statistics. random variables have lots

8 Question # 7. Independence and Identical Distributions The following is the joint probability distribution of two random variables, Y 1 and Y 2. Y Y (a) Are Y 1 and Y 2 independent? (b) Are Y 1 and Y 2 identically distributed? Are they i.i.d.? (c) Suppose we had started with 7 Democrats and 3 Republicans and dened Y i = 1 if the i-th voter chosen is a Democrat and 0 otherwise. Are we sampling with or without replacement? (d) Now suppose we had started with 7000 Democrats and 3000 Republicans and sampled without replacement. What would be the joint probability distribution of Y 1 and Y 2? Verify that in this case, the two would be approximately i.i.d..

9 Question # 8. Covariance, Correlation, Independence and Indentical Distributions The two tables below give the joint distributions of two pairs of random variables, (X, Y ) and (W, V ). W 5 15 V X 5 15 Y (a) Compute the covariance between X and Y. (b) Compute the covariance between W and V. (c) Are X and Y independent? (d) Are W and V independent? (e) Are X and Y identically distributed? Are X and Y i.i.d.? (f ) Compute the correlation between X and Y.

10 Question # 9. Covariance, Correlation, Independence and Identical Distributions The table below gives the joint distribution of two random variables, X and Y. X Y (a) Compute the covariance and correlation between X and Y. (b) Compute the conditional probability distributions for Y. These are p(y = y X = 1), p(y = y X = 0) and p(y = y X = 1). (c) Are X and Y independent? (d) What is going on? Can you reconcile your answers to (a) and (c)? Explain in your own words.

11 Question # 10. Expected Value and Variance of Linear Combinations Let R f denote the (known) return on a risk free bonds. Suppose R f = Let R 1, R 2, and R 3 denote returns on risky assets (say, stocks). (NOTE: R f is a constant and not a random variable while R 1, R 2, and R 3 are random variables). Suppose that: E(R 1 ) = 0.05, E(R 2 ) = 0.10, E(R 3 ) = 0.15, σ R1 = 0.05, σ R2 = 0.10, σ R3 = Also, suppose that R 3 is independent of R 1 and R 2. Suppose we form a portfolio by putting 40% of our wealth in riskfree bonds and 60% of our wealth in the third asset. The portfolio return is P 1 = 0.4 (.02) R 3. (a) What is the expected return on the portfolio, E(P 1 )? [Hint : c 0 = 0.008] (b) What is the variance of the portfolio return, Var(P 1 )? (c) What is the correlation between P 1 and R 3? (d) What is the correlation between P 1 and R 2? Suppose we form a second portfolio by putting 20% of our wealth in risk fee bonds, 40% in the rst asset, and 40% in the second asset, so the portfolio return is P 2 = 0.2 (.02)+0.4 R R 2. Also, suppose the correlation between R 1 and R 2 is.5. (e) What is E(P 2 )? (f ) What is Var(P 2 )? (g) What is the correlation between P 2 and R 3? Don't bother doing the math, just make an educated guess.

12 Question # 11. Expectation and Variance of Linear Combinations Suppose we have a casino game with random winnings W, where E(W ) = 0 and Var(W ) = 10. Also, if you play the game more than once, the winnings each time you play are i.i.d.. (a) Suppose that each time you play the game, the casino makes you pay a $1 cover charge. So each time you play your winnings are Y i = W i 1, where i denotes the i-th time. If you play the game three times, your winnings are Z = Y 1 + Y 2 + Y 3. What is E(Z)? What is V ar(z)? (b) Now suppose you are given the option of tripling your bet. If you triple your bet, your winnings are given by V = 3W 1. What is E(V )? What is V ar(v )? Compare playing the game once and tripling your bet versus playing three times. On average, which strategy is more protable? Which is riskier? Starting with (c) and for the rest of the question, forget about the cover charge. (c) Suppose you play the game twice. Your TOTAL winnings are T = W 1 + W 2. Your AVERAGE winnings are A = 1 2 W W 2 = 1 2 (W 1 + W 2 ). What are E(T ) and V ar(t )? What are E(A) and V ar(a)? (d) Now suppose instead of playing twice, you played n times. Again let T be your TOTAL winnings and A your AVERAGE winnings. (Your answer will depend on n.) What are E(T ), V ar(t ), E(A), and V ar(a)? (e) Now let X be i.i.d. random variables, i = 1, 2, 3,..., n, with E(X i ) = µ, V ar(x i ) = σ 2. Let X = 1 n n 1 X i. What are E(X) and V ar(x)? Your answer should be in terms of µ, σ, and n.

Random variables. Discrete random variables. Continuous random variables.

Random variables Discrete random variables. Continuous random variables. Discrete random variables. Denote a discrete random variable with X: It is a variable that takes values with some probability. Examples: