Exercises 57 and 58 refer to the following setting. In...

Exercises 57 and 58 refer to the following setting. In Exercises 14 and 18 of Section $6.1,$ we examined the probability distribution of the random variable $X=$ the amount a life insurance company earns on a 5 -year term life policy. Calculations reveal that $\mu_{X}=\$ 303.35$ and $\sigma_{X}=\$ 9707.57$ Life insurance The risk of insuring one person's life is reduced if we insure many people. Suppose that we insure two 21 -year-old males, and that their ages at death are independent. If $X_{1}$ and $X_{2}$ are the insurer's income from the two insurance policies, the insurer's average income $W$ on the two policies is $$W=\frac{X_{1}+X_{2}}{2}=0.5 X_{1}+0.5 X_{2}$$ Find the mean and standard deviation of W. (You see that the mean income is the same as for a single policy but the standard deviation is less.)

Exercises 57 and 58 refer to the following setting. In Exercises 14 and 18 of Section $6.1,$ we examined the probability distribution of the random variable $X=$ the amount a life insurance company earns on a 5 -year term life policy. Calculations reveal that $\mu_{X}=\$ 303.35$ and $\sigma_{X}=\$ 9707.57$

Life insurance The risk of insuring one person's life is reduced if we insure many people. Suppose that we insure two 21 -year-old males, and that their ages at death are independent. If $X_{1}$ and $X_{2}$ are the insurer's income from the two insurance policies, the insurer's average income $W$ on the two policies is
$$W=\frac{X_{1}+X_{2}}{2}=0.5 X_{1}+0.5 X_{2}$$
Find the mean and standard deviation of W. (You see that the mean income is the same as for a single policy but the standard deviation is less.)

Key Concepts

Scaling Property of Variance

Variance is affected by scaling a random variable: multiplying a random variable by a constant scales its variance by the square of that constant. This property is used when computing the variance (and therefore the standard deviation) of the average income, where each individual income is multiplied by 0.5.

Risk Pooling

Risk pooling in insurance refers to the reduction in overall risk, or variability of outcomes, when insuring multiple individuals. By averaging the income across multiple independent policies, the expected income remains the same while the variability (standard deviation) decreases, illustrating the benefit of diversification in risk management.

Variance of Independent Random Variables

When random variables are independent, the variance of their sum is the sum of their individual variances. This concept is crucial when determining the spread of the combined income from multiple policies, as it allows for straightforward calculation of overall risk.

Linearity of Expectation

The expectation, or mean, of a random variable is linear, meaning that the expected value of a linear combination of random variables is equal to the same linear combination of their individual expected values. This principle allows us to calculate the mean income of multiple policies by taking the average of the individual incomes.

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