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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

 {    

\dpi{150} \large 0            \dpi{150} \large \textup{otherwise}

 

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 X1 , X2 , X3 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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