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## Joint Moment Generating Function problems

## Joint Moment Generating Function example question

The joint density function of X and Y is given by

*f*(*x*, *y*) =

Find the Joint Moment Generating Function of *X* and *Y*.

Solution to this **Joint Moment Generating Function** practice problem is given in the video below!

## SOA Exam P CAS Exam 1 Expected Value Variance Standard Deviation probability problems

March 26, 2019 / 4 Comments on SOA Exam P CAS Exam 1 Expected Value Variance Standard Deviation probability problems

**Multiple Moment Generating Functions MGF example question**

A company insures homes in three cities, J, K and L. Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are

Let *X* represent the combined losses from the three cities. Calculate .

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

**Expected Value & Probability Density Function PDF example problem**

Let *X* be a continuous random variable with density function

{

Calculate the expected value of *X*.

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

**Moment Generating Function MGF example**

Let *X*_{1}* ,* *X*_{2}* , X*_{3} be a random sample from a discrete distribution with probability function

{

Determine the moment generating function, *M*(*t*), of .

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

**Expected Value & Probability Density Function PDF word problem**

An insurance company’s monthly claims are modeled by a continuous, positive random variable *X*, whose probability density function is proportional to

Determine the company’s expected monthly claims.

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

**Percentiles & Probability Density Function PDF example question**

An insurer’s annual weather related loss, *X*, is a random variable with density function

{

Calculate the difference between the 30th and 70th percentiles of *X*.

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

**Standard Deviation & Moment Generating Function MGF example problem**

An actuary determines that the claim size for a certain class of accidents is a random variable, *X*, with a moment generating function

Determine the standard deviation of the claim size for this class of accidents.

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

**Standard Deviation Table example**

A probability distribution of the claim sizes for an auto insurance policy is given in the table below:

Claim Size |
20 | 30 | 40 | 50 | 60 | 70 | 80 |

Probability |
.15 | .10 | .06 | .20 | .10 | .10 | .30 |

What percentage of the claims are within one standard deviation of the mean claim size?

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

**Variance & Cumulative Distribution Function CDF example question**

A random variable has the cumulative distribution function

{

Calculate the variance of *X*.

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

**Variance word problem**

A recent study indicates that the annual cost of maintaining and repairing a car in a town to Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car?

Solution to this Society of Actuaries **Exam P** practice problem is given in the video below!

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