Question

Insurance companies know the risk of insurance is greatly reduced if the company insures not just one person, but many people. How does this work? Let x be a random variable representing the expectation of life in years for a 25-year-old male (i.e., number of years until death). Then the mean and standard deviation of x are μ = 52.5 years and σ = 10.1 years (Vital Statistics Section of the Statistical Abstract of the United States, 116th Edition). Suppose Big Rock Insurance Company has sold life insurance policies to Joel and David. Both are 25 years old, unrelated, live in different states, and have about the same health record. Let x1 and x2 be random variables representing Joel's and David's life expectancies. It is reasonable to assume x1 and x2 are independent.

Joel, x1: 52.5; σ1 = 10.1 David, x2: 52.5; σ1 = 10.1

If life expectancy can be predicted with more accuracy, Big Rock will have less risk in its insurance business. Risk in this case is measured by σ (larger σ means more risk).

(a) The average life expectancy for Joel and David is W = 0.5x1 + 0.5x2. Compute the mean, variance, and standard deviation of W. (Use 2 decimal places.) μ σ2 σ

(b) Compare the mean life expectancy for a single policy (x1) with that for two policies (W). The mean of W is smaller. The mean of W is larger. The means are the same.

(c) Compare the standard deviation of the life expectancy for a single policy (x1) with that for two policies (W). The standard deviation of W is larger. The standard deviation of W is smaller. The standard deviations are the same.

Answer 1

Let x1 and x2 be random variables representing Joel's and David's life expectancies.

We know that the mean of x1 is

and the standard deviation of x1 is

and the mean of x2 is

and the standard deviation of x2 is

(a) The average life expectancy for Joel and David is W = 0.5x1 + 0.5x2. Compute the mean, variance, and standard deviation of W. (Use 2 decimal places.) μ σ2 σ

Given that

the expectation of W is

The variance of W is

The standard deviation of W is

ans: the mean, variance, and standard deviation of W are

(b) Compare the mean life expectancy for a single policy (x1) with that for two policies (W). The mean of W is smaller. The mean of W is larger. The means are the same.

We can see that the mean of X1 is same the mean of W which is 52.5.

ans: The means are the same

(c) Compare the standard deviation of the life expectancy for a single policy (x1) with that for two policies (W). The standard deviation of W is larger. The standard deviation of W is smaller. The standard deviations are the same.

We can see that the standard deviation of X1 is 10.1 and that of W is 7.14.

ans: The standard deviation of W is smaller.

Hence we can confirm that the risk of insurance is greatly reduced if the company insures not just one person, but many people.