An insurance policy pays a total medical benefit consisting of two parts for each claim. Let [math]X[/math] represent the part of the benefit that is paid to the surgeon, and let [math]Y[/math] represent the part that is paid to the hospital. The variance of [math]X[/math] is 5000, the variance of [math]Y[/math] is 10,000, and the variance of the total benefit, [math]X[/math] + [math]Y[/math], is 17,000. Due to increasing medical costs, the company that issues the policy decides to increase [math]X[/math] by a flat amount of 100 per claim and to increase [math]Y[/math] by 10% per claim.

Calculate the variance of the total benefit after these revisions have been made.

• 18,200
• 18,800
• 19,300
• 19,520
• 20,670

Let [math]X[/math] be a random variable that takes on the values –1, 0, and 1 with equal probabilities. Let [math]Y = X^2 [/math] . Which of the following is true?

• Cov(X, Y) > 0; the random variables X and Y are dependent
• Cov(X, Y) > 0; the random variables X and Y are independent
• Cov(X, Y) = 0; the random variables X and Y are dependent
• Cov(X, Y) = 0; the random variables X and Y are independent
• Cov(X, Y) < 0; the random variables X and Y are dependent

Let [math]X,Y[/math] be any two random variables with a joint density function. Suppose that

Which of the following statements is always true:

• [math]|\operatorname{Cov}(X,Y)| \lt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
• [math]|\operatorname{Cov}(X,Y)| \gt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
• [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math]
• If [math]\operatorname{Cov}(\operatorname{E}[X | Y],Y) = 0[/math] then [math]X [/math] and [math]Y[/math] are independent.
• If [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math] for every [math]Y[/math] then [math]X = \operatorname{E}[X | Y][/math].

You are given the following about the daily stock returns [math]r_1[/math] and [math]r_2[/math]:

• The daily stock returns [math]r_1[/math] and [math]r_2[/math] have identical marginal distributions with an expected return equal to zero.
• Given that both returns are less than -0.2, the returns are independent with a mean return of 0.05.
• Given that one of the returns is greater than -0.2, the covariance equals 0.0225 and the means equal -0.25.

Determine the covariance of [math]r_1[/math] and [math]r_2[/math].

• 0
• 0.01317
• 0.01417
• 0.0755
• 0.0795

The variables [math]X[/math] and [math]Y[/math] have joint density function

with [math]\alpha \gt 1 [/math]. Determine the limit of the covariance of [math]X[/math] and [math]Y[/math] as [math]\alpha \rightarrow 1 [/math].

• -0.0525
• -1/12
• 0
• 0.0525
• 0.1775