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]

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 distribution of the number of claims per policy during a one-year period for 10,000 insurance policies is:

   Number of Claims per Policy	Number of Policies
   0	5000
   1	5000
   2 or more	0

2. You fit a binomial model with parameters m and q using the method of maximum likelihood.

Calculate the maximum value of the loglikelihood function when [math]m = 2[/math].

• −10,397
• −7,781
• −7,750
• −6,931
• −6,730

For a group of policies, you are given:

1. Losses follow the distribution function
   [[math]] F(x) = 1-\theta/x, \, x \gt \theta [[/math]]
2. A sample of 20 losses resulted in the following:

   Interval	Number of losses
   (0,10]	9
   (10,25]	6
   (25,[math]\infty[/math])	5

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

• 5.00
• 5.50
• 5.75
• 6.00
• 6.25

You are given:

1. The random variable X has probability density function
   [[math]] f(x) = \alpha (1500)^{\alpha} (1500 + x)^{-(\alpha+1)}, \, \alpha \gt 0, \, x \gt 0[[/math]]
2. Five sample observations are:
   50 250 450 650 850

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

• 0.16
• 0.79
• 1.85
• 2.91
• 3.97

You are given the following observations on 185 small business policies:

The number of claims per policy follows a Poisson distribution with parameter [math]\lambda [/math].

Using the maximum likelihood estimate of [math]\lambda [/math] , determine the estimated probability of a policy having fewer than two claims.

• 0.79
• 0.84
• 0.89
• 0.95
• 0.98

You are given:

1. A sample of losses is: 600 700 900
2. No information is available about losses of 500 or less.
3. Losses are assumed to follow an exponential distribution with mean [math]\theta [/math].

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

• 233
• 400
• 500
• 733
• 1233

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].

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