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Practice Problems 1

Practice Problems 1 - 1 An automobile insurance company...

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1. An automobile insurance company classifies each driver as a good risk, a medium risk, or a poor risk. Of those currently insured, 30% are good risks, 50% are medium risks, and 20% are poor risks. In any given year, the probability that a driver will have a traffic accident is 0.1 for a good risk, 0.3 for a medium risk, and 0.5 for a poor risk. a) What is the probability that a randomly selected driver insured by this company had a traffic accident during 2010? b) If a randomly selected driver insured by this company had a traffic accident during 2010, what is the probability that the driver is actually a poor risk? c) If a randomly selected driver insured by this company did not have a traffic accident during 2010, what is the probability that the driver is actually a good risk? d) Suppose a driver insured by this company is not a poor risk. What is the probability that the driver had a traffic accident during 2010? e) The company announced that it will raise the insurance premiums for the drivers who either are poor risks or had a traffic accident during 2010, or both. What proportion of customers would have their premiums raised? f) Are events {a randomly selected driver is a medium risk} and {a randomly selected driver had a traffic accident during 2010} independent? g) Are events {a randomly selected driver is a medium risk} and {a randomly selected driver had a traffic accident during 2010} mutually exclusive? 2. The weight of fish in Lake Paradise follows a Normal Distribution with mean of 7.7 lbs and standard deviation of 2.5 lbs. a) What proportion of fish are between 6 lbs and 10 lbs? b) Mr. Statman boasts that he once caught a fish that was just big enough to be in the top 3% of the fish population. How much did his fish weigh? c) If one catches a fish from the bottom 9% of the population, the fish must be returned to the lake. What is the weight of the smallest fish that one can keep?

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3. Let X be a continuous random variable with the probability density function f ( x ) = k x 2 , 0 x 1, f ( x ) = 0, otherwise. a) What must the value of k be so that f ( x ) is a probability density function? b) Find the probability P( 0.4 X 0.8 ). c) Find the median of the distribution of X. d) Find μ X = E ( X ). e) Find σ X = SD ( X ). f) Find the moment-generating function of X, M X ( t ) . 4. Suppose a random variable X has the following probability density function: = - otherwise 0 1 0 ) ( x C x f x e a) What must the value of C be so that f ( x ) is a probability density function? b) Find the cumulative distribution function F ( x ) = P ( X x ) . c) Find the median of the probability distribution of X . d) Find μ X = E ( X ) . e) Find the moment-generating function of X, M X ( t ) .
5. A simple model for describing mortality in the general population in a particular country is given by the probability density function ( ) ( ) 2 6 18 100 10 252 y y y f - = , 0 < y < 100.

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