CHALLENG E A random variable with given mean and standard deviation. Here is a simple way to create a random variable X that has mean μ and standard deviation σ: X takes only the two values

μ − σ and μ + σ, each with probability 0.5. Use the definition of the mean and variance for discrete random variables to show that X does have mean μ

and standard deviation σ.

Insurance. The business of selling insurance is based on probability and the law of large numbers. Consumers buy insurance because we all face risks that are unlikely but carry high cost. Think of a fire destroying your home. So we form a group to share the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down. The insurance company sells many policies, so it can rely on the law of large numbers.

Exercises 4.87 to 4.90 explore aspects of insurance.