SOLUTION: An insurance policy is written to cover a loss, X, where X has a uniform distribution on [0, 1000]. At what level must a deductible be set in order for the expected payment to be

Question 1184022: An insurance policy is written to cover a loss, X, where X has a uniform distribution on
[0, 1000]. At what level must a deductible be set in order for the expected payment to be
25% of what it would be with no deductible?
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to determine the deductible level:
**1. Expected Payment with No Deductible:**
If there's no deductible, the expected payment is simply the expected value of the loss X. For a uniform distribution on [a, b], the expected value is (a+b)/2. In this case, a = 0 and b = 1000, so:
E[X] = (0 + 1000) / 2 = 500
**2. Desired Expected Payment:**
The problem states that the expected payment with the deductible should be 25% of the expected payment with no deductible:
Desired Expected Payment = 0.25 * 500 = 125
**3. Expected Payment with a Deductible:**
Let 'd' be the deductible. The payment Y made by the insurance company is:
* Y = 0, if X ≤ d
* Y = X - d, if X > d
The expected payment with the deductible is:
E[Y] = ∫ (x - d) * f(x) dx (integrated from d to 1000)
Where f(x) is the probability density function of the uniform distribution, which is 1/(1000 - 0) = 1/1000 for 0 ≤ x ≤ 1000.
So,
E[Y] = ∫ (x - d) * (1/1000) dx (integrated from d to 1000)
E[Y] = (1/1000) * [x²/2 - dx] (evaluated from d to 1000)
E[Y] = (1/1000) * [(1000²/2 - 1000d) - (d²/2 - d²)]
E[Y] = (1/1000) * [500000 - 1000d - d²/2 + d²]
E[Y] = (1/1000) * [500000 - 1000d + d²/2]
E[Y] = 500 - d + d²/2000
**4. Solve for the Deductible (d):**
We want E[Y] to equal 125:
125 = 500 - d + d²/2000
0 = 375 - d + d²/2000
Multiply by 2000 to get rid of the fraction:
0 = 750000 - 2000d + d²
This is a quadratic equation. Rearrange it into standard form:
d² - 2000d + 750000 = 0
Use the quadratic formula to solve for d:
d = (-b ± √(b² - 4ac)) / 2a
d = (2000 ± √((-2000)² - 4 * 1 * 750000)) / 2
d = (2000 ± √(4000000 - 3000000)) / 2
d = (2000 ± √1000000) / 2
d = (2000 ± 1000) / 2
This gives us two possible solutions for d:
* d = (2000 + 1000) / 2 = 1500
* d = (2000 - 1000) / 2 = 500
Since the deductible cannot be greater than the maximum possible loss (1000), the correct solution is:
d = 500
**Answer:** The deductible must be set at 500.

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