You are given:

1. Losses follow a single-parameter Pareto distribution with density function:
   [[math]]f(x) = \frac{\alpha}{x^{\alpha+1}}, \, x\gt1, \, 0 \lt \alpha \lt \infty [[/math]]
2. A random sample of size five produced three losses with values 3, 6 and 14, and two losses exceeding 25.

Calculate the maximum likelihood estimate of [math]\alpha [/math]

• 0.25
• 0.30
• 0.34
• 0.38
• 0.42

You are given:

1. Low-hazard risks have an exponential claim size distribution with mean [math]\theta[/math].
2. Medium-hazard risks have an exponential claim size distribution with mean [math]2 \theta [/math].
3. High-hazard risks have an exponential claim size distribution with mean [math]3 \theta [/math] .
4. No claims from low-hazard risks are observed.
5. Three claims from medium-hazard risks are observed, of sizes 1, 2 and 3.
6. One claim from a high-hazard risk is observed, of size 15.

Calculate the maximum likelihood estimate of [math]\theta[/math].

• 1
• 2
• 3
• 4
• 5

The number of claims follows a negative binomial distribution with parameters [math]\beta[/math] and r, where [math]\beta[/math] is unknown and r is known. You wish to estimate [math]\beta[/math] based on [math]n[/math] observations, where [math]x[/math] is the mean of these observations.

Determine the maximum likelihood estimate of [math]\beta[/math] .

• [math]\frac{\overline{x}}{r^2}[/math]
• [math]\frac{\overline{x}}{r}[/math]
• [math]\overline{x}[/math]
• [math]r\overline{x}[/math]
• [math]r^2\overline{x}[/math]

You are given:

1. Losses follow an exponential distribution with mean [math]\theta[/math] .
2. A random sample of 20 losses is distributed as follows:

Calculate the maximum likelihood estimate of [math]\theta[/math].

• Less than 1950
• At least 1950, but less than 2100
• At least 2100, but less than 2250
• At least 2250, but less than 2400
• At least 2400

You observe the following five ground-up claims from a data set that is truncated from below at 100:

125   150   165   175   250

You fit a ground-up exponential distribution using maximum likelihood estimation.

Calculate the mean of the fitted distribution.

• 73
• 100
• 125
• 156
• 173

Losses come from a mixture of an exponential distribution with mean 100 with probability p and an exponential distribution with mean 10,000 with probability 1 − p.

Losses of 100 and 2000 are observed.

Determine the likelihood function of p.

• [math]\left (\frac{pe^{-1}}{100} \frac{(1-p)e^{-0.01}}{10,000}\right) \left( \frac{pe^{-20}}{100}\frac{(1-p)e^{-0.2}}{10,000}\right)[/math]
• [math]\left (\frac{pe^{-1}}{100} \frac{(1-p)e^{-0.01}}{10,000}\right) + \left( \frac{pe^{-20}}{100}\frac{(1-p)e^{-0.2}}{10,000}\right)[/math]
• [math]\left (\frac{pe^{-1}}{100}+ \frac{(1-p)e^{-0.01}}{10,000}\right) \left( \frac{pe^{-20}}{100} + \frac{(1-p)e^{-0.2}}{10,000}\right)[/math]
• [math]\left (\frac{pe^{-1}}{100} +\frac{(1-p)e^{-0.01}}{10,000}\right) + \left( \frac{pe^{-20}}{100}+\frac{(1-p)e^{-0.2}}{10,000}\right)[/math]
• [math]p\left (\frac{pe^{-1}}{100} +\frac{(1-p)e^{-0.01}}{10,000}\right) + (1-p)\left( \frac{pe^{-20}}{100}+\frac{(1-p)e^{-0.2}}{10,000}\right)[/math]

You are given the following three observations:

0.74  0.81  0.95

You fit a distribution with the following density function to the data:

Calculate the maximum likelihood estimate of [math]p[/math].

• 4.0
• 4.1
• 4.2
• 4.3
• 4.4

You are given:

1. The number of claims follows a Poisson distribution with mean [math]\lambda [/math] .
2. Observations other than 0 and 1 have been deleted from the data.
3. The data contain an equal number of observations of 0 and 1.

Calculate the maximum likelihood estimate of [math]\lambda [/math] .

• 0.50
• 0.75
• 1.00
• 1.25
• 1.50

You are given the following 20 bodily injury losses before the deductible is applied:

Past experience indicates that these losses follow a Pareto distribution with parameters [math]\alpha [/math] and [math]\theta = 10,000 [/math].

Calculate the maximum likelihood estimate of [math]\alpha [/math].

• Less than 2.0
• At least 2.0, but less than 3.0
• At least 3.0, but less than 4.0
• At least 4.0, but less than 5.0
• At least 5.0

Personal auto property damage claims in a certain region are known to follow the Weibull distribution:

A sample of four claims is:

130  240  300  540

The values of two additional claims are known to exceed 1000.

Calculate the maximum likelihood estimate of [math]\theta[/math].

• Less than 300
• At least 300, but less than 1200
• At least 1200, but less than 2100
• At least 2100, but less than 3000
• At least 3000