To calculate the expected value, we use the formula for expected value $E(X)$ of a discrete random variable:

$E(X) = \sum \left( \text{Amount of Claim} \times \text{Probability} \right)$

Given the dataTo calculate the expected value, we use the formula for expected value $E(X)$ of a discrete random variable:

$E(X) = \sum \left( \text{Amount of Claim} \times \text{Probability} \right)$

Given the

• $\$0$ with probability $0.70$
• $\$20,000$ with probability $0.18$
• $\$40,000$ with probability $0.08$
• $\$60,000$ with probability $0.02$
• $\$80,000$ with probability $0.01$
• $\$100,000$ with probability $0.01$

Let's calculate the expected value:

$E(X) = (0 \times 0.70) + (20,000 \times 0.18) + (40,000 \times 0.08) + (60,000 \times 0.02) + (80,000 \times 0.01) + (100,000 \times 0.01)$

I'll compute this now.# Defining the amounts and their corresponding probabilities amounts = [0, 20000, 40000, 60000, 80000, 100000] probabilities = [0.70, 0.18, 0.08, 0.02, 0.01, 0.01]

Calculating the expected value

expected_value = sum(amount * prob for amount, prob in zip(amounts, probabilities)) expected_valueThe expected value of the claims is $9,800.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. How does changing the probability of a $40,000 claim affect the expected value?
2. What is the variance of this distribution?
3. What would happen to the expected value if the probability of a $0 claim decreased?
4. How would you compute the standard deviation for this set of claims?
5. If the probability of a $100,000 claim increased, how would that affect the risk to the insurance company?

Tip: The expected value is a measure of the center or "average" of a probability distribution. It's useful in assessing long-term outcomes in insurance and other probabilistic situations.