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

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

• -56.25
• 0
• 18.75
• 68.75
• 168.75

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

An industrial company provides health insurance to employees located at four different plants. Health insurance costs at each plant are independent of the costs at any other plant. Plant managers have calculated the following statistics:

Calculate the standard deviation of total company health insurance costs.

• 1.4
• 2.1
• 2.9
• 5.5
• 6.3

Profit for a new product is given by [math]Z = 3X-Y-5[/math]. [math]X[/math] and [math]Y[/math] are independent random variables with [math]\operatorname{Var}(X) = 1 [/math] and [math]\operatorname{Var}(Y) = 2[/math].

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

• 1
• 5
• 7
• 11
• 16

A joint density function is given by

where [math]k[/math] is a constant. Calculate [math]\operatorname{Cov}(X,Y)[/math].

• -1/6
• 0
• 1/9
• 1/6
• 2/3

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

An employer provides disability benefits to its employees for work-related and other injuries. The random variables [math]X[/math] and [math]Y[/math] denote the employer’s annual expenditures for work-related and other injuries, respectively. An actuarial study reveals the following information about [math]X[/math] and [math]Y[/math]:

1. The density of [math]X[/math] is [math]f(x)= \frac{1}{20 \sqrt{5}} e^{-x/(20 \sqrt{5})}, \, x \gt 0[/math]
2. [math]\operatorname{Var}(Y)=12500 [/math]
3. The correlation between [math]X[/math] and [math]Y[/math] is 0.20.

Calculate the variance of the employer’s total expenditures for work-related and other injuries.

• 12,500
• 13,500
• 15,500
• 16,500
• 18,972