The statewide auto insurance company developed the following probability distribution for automobile collision

Question

claims paid during the past year...

Payment 0, 500, 1000,
2000, 5000, 8000, 10,000.

Probability .83, .06, .05, .02, .02, .01, .01
a. set up intervals of random numbers that can be used to generate automobile collision claim payments
b. using the numbers (.9850, .2122, .5590, .0004, .5177, .0064, .2044, .2390, .7305, .6351, .4971, .9569, .1091, .9907, .1900, .9403, .3545, .8083, .4704, .5274. )simulate payments for 20 policyholders. How many claims are paid and what is the total amount paid to the policyholders

Answer 1

Dear cherita,

a. You want your intervals such that a uniform random number between 0 and 1 will give you each of your seven outcomes with the stated probabilities. Thus, you need to divide the interval between 0 and 1 proportionally. This is easy to do just by cumulating the probabilities (i.e., adding them sequentially to determine where each sub-interval begins and ends).

Starting from 0 (the lower bounds of the first interval), add 0.83, which gives 0.83 as the upper bounds of the first sub-interval. This also serves as the lower bounds of the next sub-interval. Next, add 0.06, which gives 0.83 + 0.06 = 0.89 as the upper bounds of the second sub-interval (and the lower bounds for the third sub-interval). By continuing this procedure, you will have obtained the intervals you desire, mapped on the interval (0,1):

0.00 0.83 sub-interval 1 (claims amount 0),
0.83 0.89 sub-interval 2 (claims amount 500),
0.89 0.94 sub-interval 3 (claims amount 1000),
0.94 0.96 sub-interval 4 (claims amount 2000),
0.96 0.98 sub-interval 5 (claims amount 5000),
0.98 0.99 sub-interval 6 (claims amount 8000), and
0.99 1.00 sub-interval 7 (claims amount 10000).

Thus, when you draw a random number U, you determine in which of the possible sub-interval it falls, and that will correspond to a particular claims amount. For example, if U = 0.9765, you would see that 0.96 < U <= 0.98, so U would be in sub-interval 5, corresponding to a claim of 5000.


b. I just gave an example how to simulate any uniform random draw in the unit interval (0,1), so all you have to do is make a table using the list of numbers you are given, looking up the corresponding claim amounts for each. You can then answer questions on the total number of claims and payments to policy holders. Note that when 0.00 < U <= 0.83, there is no claim paid.

Answer 2

1

Answer 3

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