Two loss variables [math]L_1[/math] and [math]L_2[/math] have a strictly increasing continuous joint cumulative distribution [math]F(x,y)[/math]. Which of the following expressions represent the probability that both losses exceed their 95^th percentile?

• [math]F(F_X^{-1}(0.95), F_Y^{-1}(0.95))[/math] with [math]F_X(x) = F(x,\infty)[/math] and [math]F_Y(y) = F(\infty, y)[/math]
• [math]F(F_X^{-1}(0.95), F_Y^{-1}(0.95)) - 0.9.[/math] with [math]F_X(x) = F(x,\infty)[/math] and [math]F_Y(y) = F(\infty, y)[/math]
• [math]F(F_X(0.95), F_Y(0.95)) - 0.9.[/math] with [math]F_X(x) = F(x,\infty)[/math] and [math]F_Y(y) = F(\infty, y)[/math]
• [math]0.05^2[/math]
• [math]F(F_X^{-1}(0.95), F_Y^{-1}(0.95)) - 0.9.[/math] with [math]F_X(x) = F(\infty, x)[/math] and [math]F_Y(y) = F(y, \infty)[/math]

The joint distribution of claim frequency and claim severity, conditional on the claim frequency being positive, is given below:

If the probability that the claim frequency is zero is 0.75, determine the variance of the claim frequency.

• 0.2475
• 0.5123
• 0.6625
• 0.8833
• 1

Let [math]T_1[/math] be the time between a car accident and reporting a claim to the insurance company. Let [math]T_2[/math] be the time between the report of the claim and payment of the claim. The joint density function of [math]T_1[/math] and [math]T_2[/math], [math]f(t_1,t_2)[/math], is constant over the region

and zero otherwise.

Calculate [math]\operatorname{E}(T_1 + T_2)[/math], the expected time between a car accident and payment of the claim.

• 4.9
• 5.0
• 5.7
• 6.0
• 6.7

A family buys two policies from the same insurance company. Losses under the two policies are independent and have continuous uniform distributions on the interval from 0 to 10. One policy has a deductible of 1 and the other has a deductible of 2. The family experiences exactly one loss under each policy.

Calculate the probability that the total benefit paid to the family does not exceed 5.

• 0.13
• 0.25
• 0.30
• 0.32
• 0.42

The joint density function of [math]\ln(X)[/math] and [math]\ln(Y)[/math] equals [math]f[/math]. Which of the following expressions represent the joint density function of [math]X[/math] and [math]Y[/math] evaluated at [math](x,y)[/math]?

• [math]y^{-1}x^{-1}f(\ln(x),\ln(y))[/math]
• [math]f(\ln(x),\ln(y))[/math]
• [math]y^{-1}x^{-1}f(x,y)[/math]
• [math]e^x e^yf(e^x,e^y)[/math]
• [math]x^{-1}f(\ln(x),x) + y^{-1}f(x,\ln(y))[/math]