The variables [math]S[/math] and [math]T[/math] have the joint density function

Determine [math]\operatorname{Cov}(T,S)[/math].

1. -56.25
2. 0
3. 18.75
4. 68.75
5. 168.75

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

Let [math]X,Y[/math] be any two random variables. Which of the following statements is always true:

1. [math]|\operatorname{Cov}(X,Y)| \lt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
2. [math]|\operatorname{Cov}(X,Y)| \gt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
3. [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math]
4. If [math]\operatorname{Cov}(\operatorname{E}[X | Y],Y) = 0[/math] then [math]X [/math] and [math]Y[/math] are independent.
5. 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].

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

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

• 0
• 0.01317
• 0.01417
• 0.0755
• 0.0795

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[/math] and [math]Y[/math] be continuous random variables with joint density function

Calculate the covariance of [math]X[/math] and [math]Y[/math].

• 0.04
• 0.25
• 0.67
• 0.80
• 1.24

[math]X[/math] and [math]Y[/math] denote the values of two stocks at the end of a five-year period. [math]X[/math] is uniformly distributed on the interval (0, 12). Given [math]X = x[/math], [math]Y[/math] is uniformly distributed on the interval [math](0, x)[/math].

Calculate [math]\operatorname{Cov}(X, Y)[/math] according to this model.

• 0
• 4
• 6
• 12
• 24

Let [math]X[/math] denote the size of a surgical claim and let [math]Y[/math] denote the size of the associated hospital claim. An actuary is using a model in which

• [math]\operatorname{E}[X] = 5 [/math]
• [math]\operatorname{E}[X^2] = 27.4 [/math]
• [math]\operatorname{E}[Y] = 7 [/math]
• [math]\operatorname{E}[Y^2] = 51.4 [/math]
• [math]\operatorname{Var}[X + Y] = 8 [/math]

Let [math]C_1 = X + Y [/math] denote the size of the combined claims before the application of a 20% surcharge on the hospital portion of the claim, and let [math]C_2[/math] denote the size of the combined claims after the application of that surcharge.

Calculate [math]\operatorname{Cov}(C_1,C_2)[/math] .

• 8.80
• 9.60
• 9.76
• 11.52
• 12.32

An actuary analyzes a company’s annual personal auto claims, [math]M[/math], and annual commercial auto claims, [math]N[/math]. The analysis reveals that [math]\operatorname{\operatorname{Var}}(M) = 1600 [/math], [math]\operatorname{Var}(N) = 900 [/math], and the correlation between [math]M[/math] and [math]N[/math] is 0.64.

Calculate [math]\operatorname{\operatorname{Var}}(M + N)[/math].

• 768
• 2500
• 3268
• 4036
• 4420

Points scored by a game participant can be modeled by [math]Z = 3X + 2Y – 5[/math]. [math]X[/math] and [math]Y[/math] are independent random variables with [math]\operatorname{Var} (X) = 3[/math] and [math]\operatorname{Var} (Y) = 4[/math].

Calculate [math]\operatorname{Var}(Z) [/math].

• 12
• 17
• 38
• 43
• 68