Question

5.1 basics of probability distributions. compute the mean and standard deviation of a discrete random variable. based on historical data, an insurance company estimates that a particular customer has a 2.4% likelihood of having an accident in the next year, with the average insurance payout being $2300. if the company charges this customer an annual premium of $130, what is the companys expected value of this insurance policy?

Explanation:

Step1: Define the two - outcome cases

Let's consider two cases for the insurance company. Case 1: the customer has an accident, and case 2: the customer does not have an accident.

Step2: Calculate the probability and profit for the accident case

The probability of an accident $p = 0.024$. The cost to the company in case of an accident is the payout of $\$2300$, and the company receives a premium of $\$130$. So the profit in case of an accident $x_1=130 - 2300=- 2170$.

Step3: Calculate the probability and profit for the non - accident case

The probability of no accident $q=1 - p=1 - 0.024 = 0.976$. The company just receives the premium of $\$130$, so the profit in case of no accident $x_2 = 130$.

Step4: Use the expected - value formula

The formula for the expected value $E(X)$ of a discrete random variable is $E(X)=p\times x_1+q\times x_2$. Substitute $p = 0.024$, $x_1=-2170$, $q = 0.976$, and $x_2 = 130$ into the formula:

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Answer:

$74.8$