JSB381 Lecture Notes - Normal Distribution, Fair Coin, Cumulative Distribution Function
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A salesperson makes four calls per year. A sample of 100 days given the following frequencies of sales volumes.
Number of Sales | Observed Frequency (days) |
0 | 30 |
1 | 32 |
2 | 25 |
3 | 10 |
4 | 3 |
TOTAL | 100 |
Records show sales are made to 30% of all sales calls. Assuming independent sales calls, the number of sales per day should follow a binomial distribution. The Binomial distribution is represented by:
For this exercise, assume that the population has a binomial probability distribution with n=4, p=0.30 and x= 0, 1, 2, 3 and 4.
(a) Compute the expected frequencies for x=0, 1, 2, 3 and 4 by using the binomial probability function. Combine categories if necessary to satisfy the requirement that the expected frequency is five or more for all categories.
(b) Use the goodness of fit test to determine whether the assumption of a binomial probability distribution should be rejected. Because no parameters of the Binomial probability distribution were estimated from the sample data, the degrees of freedom are k-1 .
Consider a car owner who has an 80% chance of no accidents in a year. For simplicity, assume that there is a 10% probability that after the accident the car will need repairs costing $500, an 8% probability that the repairs will cost $5,000, and a 2% probability that the car will need to be replaced, which will cost
$15,000. Based on this information, the probability distribution, f(x), of the random variable, X, loss due to accident, is:
f(x) = |
0.8 x = $0 |
0.10 x = $500 |
0.08 x = $5,000 |
0.02 x = $15,000 |
where the first column is the probability of the event (i.e. P(x=0) =0.8) and the second is the severity.
Assuming risk retention, calculate the object risk to the car owner. Show ALL your work.
Now consider an insurance company that will reimburse repair costs resulting from accidents for 100 car owners, each with the same probabilities and losses as in part a). Calculate the objective risk for the insurance company. How is this number compared to that of part a)? Explain. Show ALL your work.
Suppose that the insurance company provides insurance to the same 100 car owners but now it introduces a deductible of $500. The claim payment distribution for EACH policy would now be:
f(y)= |
0.90 x = $0 or $500 y = $0 |
0.08 x = $5,000 y = $4,500 |
0.02 x = $15,000 y = $14,500 |
where