Question 1185234: An insurance company charges a customer an annual premium of $100, and
there is a probability of 0.9 that the customer will not need to make a claim. If the
customer does make a claim, the amount of the claim $𝑋 has a probability density
function
𝑓(𝑥) = 𝑥(1800 − 𝑥)/972,000,000
For 0 ≤ 𝑥 ≤ 1800. Each customer also incurs administrative costs to the insurance company
of $5. If the insurance company has 10,000 customers, what is its expected annual profit?
Would you expect the customers’ claim to be independent of each other?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the insurance company's expected annual profit:
**1. Calculate the expected claim amount:**
The expected claim amount is calculated by integrating the product of the claim amount (x) and the probability density function f(x) over the range of possible claim amounts:
E[X] = ∫₀¹⁸⁰⁰ x * f(x) dx
E[X] = ∫₀¹⁸⁰⁰ x * [x(1800 - x) / 972,000,000] dx
E[X] = (1/972,000,000) ∫₀¹⁸⁰⁰ (1800x² - x³) dx
E[X] = (1/972,000,000) [600x³ - (x⁴/4)] from 0 to 1800
E[X] = (1/972,000,000) * [600(1800)³ - (1800)⁴/4]
E[X] = (1/972,000,000) * 1,944,000,000
E[X] = $1000
**2. Calculate the expected profit per customer:**
* Premium per customer: $100
* Administrative cost per customer: $5
* Probability of making a claim: 1 - 0.9 = 0.1
* Expected claim amount: $1000
Expected profit per customer = Premium - Administrative cost - (Probability of claim * Expected claim amount)
Expected profit per customer = $100 - $5 - (0.1 * $1000)
Expected profit per customer = $95 - $100
Expected profit per customer = -$5
**3. Calculate the expected annual profit for 10,000 customers:**
Expected annual profit = Expected profit per customer * Number of customers
Expected annual profit = -$5 * 10,000
Expected annual profit = -$50,000
**4. Independence of claims:**
It's reasonable to assume that customers' claims are independent of each other. One customer making a claim should not generally affect the probability of another customer making a claim. There might be some rare exceptions (e.g., a widespread natural disaster causing many claims), but for the vast majority of individual claims, independence is a valid assumption.
**Conclusion:**
The insurance company's expected annual profit is -$50,000. This means that, on average, the company can expect to lose $50,000 per year with the current premium structure. The assumption of independence between customer claims is generally reasonable.
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