Question 1185234
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.