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

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

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

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

The random variable [math]X[/math] has survival function:

Two values of [math]X[/math] are observed to be 2 and 4. One other value exceeds 4.

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

• Less than 4.0
• At least 4.0, but less than 4.5
• At least 4.5, but less than 5.0
• At least 5.0, but less than 5.5
• At least 5.5

You are given:

1. At time 4 hours, there are 5 working light bulbs.
2. The 5 bulbs are observed for [math]p[/math] more hours.
3. Three light bulbs burn out at times 5, 9, and 13 hours, while the remaining light bulbs are still working at time 4 + [math]p[/math] hours.
4. The distribution of failure times is uniform on (0, [math]\omega[/math] ) .
5. The maximum likelihood estimate of [math]\omega [/math] is 29.

Calculate [math]p[/math].

• Less than 10
• At least 10, but less than 12
• At least 12, but less than 14
• At least 14, but less than 16
• At least 16