Quant Ch 10

Drivers for a freight company have a varying number of delivery stops. The mean number of stops is 6 with standard deviation 2. Two drivers operate independently of one another. (A) Identify the two random variables and summarize your assumptions. (B) What is the mean and standard deviation of the number of stops made in a day by these two​ drivers? (E) It is more likely the case that the driver who spends more time making deliveries also has fewer to​ make, and conversely that the other driver has more delivers to make. Does this suggest that the two random variables may not meet the assumptions of the​ problem?

(A) Assume that the covariance between the number of delivery stops is zero. (B)E ( X1+X2​)=12 ​SD(X1+X2​)=2.83 (E) Yes. It suggests that the counts of the number of deliveries are negatively correlated and not independent.

Two classmates enjoy playing online poker. They both claim to win ​$200 on average when they play for an​ evening, even though they play at different sites on the Web. They do not always win the same​ amounts, and the SD of the amounts won is ​$125. (A) Identify the two random variables and summarize your assumptions. Let X1 and X2 denote the winnings of the first and second​ classmates, respectively. What are the appropriate​ assumptions? (B) The classmates have decided to play in a tournament and are seated at the same virtual game table. How will this affect your assumptions about the random​ variables?

(A) Assume that the​ classmates' daily winnings are independent and that the classmates winnings are randomly distributed. (B) X1and X2 can no longer reasonably be assumed to be independent because if one classmate wins the most money at the​ table, then the other classmate cannot win the most money at the table.

An office complex leases space to various companies. These leases include energy costs associated with heating during the winter. To anticipate costs in the coming​ year, the managers developed two random variables X and Y to describe costs for equivalent amounts of heating oil​ (X) and natural gas​ (Y) in the coming year. Both X and Y are measured in dollars per Btu of heat produced. The complex uses both fuels for​ heating, with YσX=σY. (A) If managers believe that costs for both fuels tend to rise and fall​ together, then they should model X and Y as independent. (B) Because the means and SDs of these random variables are the​ same, the random variables X and Y are identically distributed. (C) If told that the costs of heating oil and natural gas are​ uncorrelated, an analyst should then treat the joint distribution as ​p(x,y)=​p(x)p(y).

(A) The statement is false. If the costs move​ simultaneously, they should be treated as dependent random variables. (B) False, equality of mean and variance does not imply that ​p(x)=​p(y). (C) The covariance can equal 0 without the two variables being independent. The joint distribution ​p(x,y)=​p(x)p(y) can only be used when the random variables are independent. (​False, zero covariance does not imply independence.)

A pharmaceutical company has developed a new drug that helps insomniacs sleep. In tests of the​ drug, it records the daily number of hours asleep and awake. Let X denote the number of hours awake and let Y denote the number of hours asleep for a typical patient. (A) Explain why the company must model X and Y as dependent random variables. (B) If the company considers the difference between the number of hours awake and the number​ asleep, how will the dependence affect the SD of this​ comparison?

(A) X and Y must be dependent because X+Y=24. (B) The standard deviation of the difference will be higher because the covariance of X and Y is negative.

Kitchen remodeling is a popular way to improve the value of a home. (A) If X denotes the amount spent for labor and Y the cost for new​ appliances, do you think these would be positively​ correlated, negatively​ correlated, or​ independent? (B) If a family is on a strict budget that limits the amount spent on remodeling to​ $25,000, does this change your impression of the dependence between X and​ Y?

(A) X and Y would likely be positively correlated because each new appliance would require labor to install it. (B) A budget constraint might produce a negative association because the family cannot afford to spend a lot on both labor and new appliances.

A student budgets​ $60 weekly for gas and a few quick meals​ off-campus. Let X denote the amount spent for gas and Y the amount spent for quick meals in a typical week. Assume this student is very disciplined and sticks to the​ budget, spending​ $60 on these two things each week. (A) Can we model X and Y as independent random​ variables? Explain. (B) Suppose we assume X and Y are dependent. What is the effect of this dependence on the variance of X+​Y?

(A) ​No, X and Y are dependent​ because, for​ example, one can write Y=60−X. (B) Since X+Y=​60, the variance is 0.

As a baseline when planning future​ advertising, retail executives treat the dollar values of sales on consecutive weekends as independent and identically distributed​ (iid) random variables. The amounts sold on two consecutive weekends​ (call these Upper X1 and Upper X2​) are iid random variables with mean muμ and standard deviation sigmaσ. (A) On​ average, retailers expect to sell the same amount on the first weekend and the second weekend.

(A)True

If investors want portfolios with small risk​ (variance), should they look for investments that have positive​ covariance, have negative​ covariance, or are​ uncorrelated?

Negative The addition rule for variances of sums is shown​ below, where X and Y are random variables and​ Cov(X,Y) is the covariance between X and Y. If the investors want to decrease the variance of the​ return, they should pick investments that have a negative covariance. ​Var(X+​Y)=​Var(X)+​Var(Y)+​2Cov(X,Y)

If the covariance between the prices of two investments is 800,000​, does this tell you that the correlation between the two is close to​ 1?

No, it only tells us that the correlation is positive.

Sharpe ratio of a random variable

S(Y)

The addition rule for covariance is given by ​Var(X+​Y)=​Var(X)+​Var(Y)+​2Cov(X,Y). Notice that only when the covariance is zero will ​Var(X+​Y)=​Var(X)+​Var(Y). Carefully look over the choices to find the notation corresponding to positive covariance

The addition rule for covariance is given by ​Var(X+​Y)=​Var(X)+​Var(Y)+​2Cov(X,Y). If the covariance is​ positive, ​Var(X+​Y) must be greater than ​Var(X)+​Var(Y).

What's the covariance between a random variable X and a​ constant? The covariance is equal to _______

The covariance between random variables is the expected value of the product of deviations from the means. The formula for the covariance is shown below. Cov(X, Y)=E((X−μX)(Y−μY)) The covariance between a random variable X and a constant is zero because a constant does not deviate from its mean.​ Thus, the expected value of the product of deviations from the means is​ zero, as shown below. 0

What's the covariance between a random variable X and a​ constant?

The covariance is equal to 0

Identify the symbol for a joint probability distribution

The probability distribution of X is denoted​ p(x). The probability distribution of Y is denoted​ p(y). The joint probability distribution of X and Y is denoted as​ p(x,y).

What notation implies that X and Y are identically​ distributed?

The probability distribution of X is denoted​ p(x). The probability distribution of Y is denoted​ p(y). This notation p(x)=p(y) implies that X and Y are identically distributed.

What is true for uncorrelated random​ variables?

The variance of the sum of two random variables is the sum of their variances plus twice their covariance. If X and Y are​ uncorrelated, Var(X+Y)=VAR(X)+VAR(Y). The notation Var(X+Y)=VAR(X)+VAR(Y) is true for uncorrelated random variables.