Year 1 ground-up losses have an exponential distribution with mean $1,250. Inflation of 5% impacts all claims from year 1 to year 2. The policy has a deductible of $400 in effect during years 1 and 2. If x₁ denotes the probability that non-zero payments exceed $500 in year 1 and x₂ denotes the probability that non-zero payments exceed $500 in year 2, determine the interval containing the ratio x₂/x₁.

• [0.5, 0.7]
• [0.98, 1.03]
• [1.1, 1.2]
• [1.3, 1.4]
• 1.5+

The time, [math]T[/math], that a manufacturing system is out of operation has cumulative distribution function

The resulting cost to the company is [math]Y = T^2[/math]. Let [math]g[/math] be the density function for [math]Y[/math]. Determine [math]g(y)[/math], for [math]y \gt 4 [/math].

• [math]\frac{4}{y^2}[/math]
• [math]\frac{8}{y^{3/2}}[/math]
• [math]\frac{8}{y^3}[/math]
• [math]\frac{16}{y}[/math]
• [math]\frac{1024}{y^5}[/math]

An investment account earns an annual interest rate [math]R[/math] that follows a uniform distribution on the interval (0.04, 0.08). The value of a 10,000 initial investment in this account after one year is given by [math]V = 10 000e^R. [/math] Let [math]F[/math] be the cumulative distribution function of [math]V[/math]. Determine [math]F(v)[/math] for values of [math]v[/math] that satisfy [math]0 \lt F(v) \lt 1 [/math].

• [math]\frac{10000e^{v /10000} − 10408}{425}[/math]
• [math]25e^{v/10000} - 0.04[/math]
• [math]\frac{v-10408}{10833-10408}[/math]
• [math]\frac{25}{v}[/math]
• [math]25\left[ \ln(v/10000) - 0.04 \right][/math]

An actuary models the lifetime of a device using the random variable [math]Y = 10X^{0.8}[/math], where [math]X[/math] is an exponential random variable with mean 1. Let [math]f(y)[/math] be the density function for [math]Y[/math].

Determine [math]f(y)[/math] for [math]y\gt0[/math].

• [math]10 y^{0.8} \exp(−8 y^{−0.2} )[/math]
• [math]8 y^{−0.2} \exp(−10 y^{0.8} )[/math]
• [math]8 y^{−0.2} \exp[−(0.1y)^{1.25} ][/math]
• [math](0.1y )^{1.25} \exp[−0.125(0.1y)^{0.25} ][/math]
• [math]0.125(0.1y )^{0.25} \exp[−(0.1y )^{1.25} ][/math]

Let [math]T[/math] denote the time in minutes for a customer service representative to respond to 10 telephone inquiries. [math]T[/math] is uniformly distributed on the interval with endpoints 8 minutes and 12 minutes. Let [math]R[/math] denote the average rate, in customers per minute, at which the representative responds to inquiries, and let [math]f(r)[/math] be the density function for [math]R[/math].

Determine [math]f(r)[/math], for [math]\frac{10}{12} \lt r \lt \frac{10}{8}[/math].

• 12/5
• 3-5/2r
• [math]3r-5\ln(r)/2[/math]
• [math]\frac{10}{r^2}[/math]
• [math]\frac{5}{2r^2}[/math]

A number [math]U[/math] is chosen at random in the interval [math][0,1][/math]. Find the probability that [math]T = U/(1 - U) \lt 1/4[/math].

• 1/6
• 1/5
• 1/4
• 1/2
• 2/3

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

An investor has invested $50,000 in a stock. Denote by [math]S_t[/math] the stock price at time [math]t[/math] expressed in years with [math]S_0[/math] the current stock price. Suppose [math]\ln(S_t/S_0) [/math] is normally distributed with mean [math]\mu = 0.05t [/math] and standard deviation [math]\sigma = 0.1\sqrt{t} [/math]. Determine the interval containing the 99^th percentile for the investor's $50,000 investment for a 30 day period. Assume 365 days in a year

• [46000, 50000]
• [52000, 55000]
• [58000, 60000]
• [75000, 77000]
• [77000, [math]\infty[/math]]

Losses for year 1 equal

with [math]X[/math] a non-negative random variable bounded by 1 with cumulative distribution function

The annual inflation rate for year 1 has the following discrete distribution: 50% probability of 2% inflation, 30% probability of 1% inflation and 20% probability of no inflation.

Assuming that inflation is independent of loss, determine the interval containing the probability that losses for year 2 exceed $2,000.

• [0.035, 0.04]
• [0.045, 0.05]
• [0.055, 0.06]
• [0.07, 0.08]
• [0.09, 0.1]

The monthly profit of Company I can be modeled by a continuous random variable with density function [math]f[/math]. Company II has a monthly profit that is twice that of Company I. Let [math]g[/math] be the density function for the distribution of the monthly profit of Company II.

Determine [math]g(y)[/math] where it is not zero.

• [math]\frac{1}{2}f(\frac{y}{2})[/math]
• [math]f(\frac{y}{2})[/math]
• [math]2f(\frac{y}{2})[/math]
• [math]2f(y)[/math]
• [math]2f(2y)[/math]

Suppose that [math]Y = g(X)[/math] has a cumulative distribution function

with [math]F(x)[/math] the distribution function of [math]X[/math]. Which of the following functions equal [math]g[/math]?

• [math]\ln(\ln(x))[/math]
• [math]\ln(x)[/math]
• [math]x-e[/math]
• [math](x-e)^2[/math]
• Impossible to tell