Applications of Mathematical Expectation

Applications of Mathematical Expectation

Applications in Decision Making

Application 1: maximizing expected return Under uncertainty, a decision maker usually chooses the alternative with the largest expected return.

Application 2: risk assessment Risk is assessed by computing the expected loss.

Applications in Statistics

Application 3: the sample mean The sample mean is an unbiased estimator of the population expectation.

Application 4: parameter estimation Many statistical methods are based on the properties of expectation.

Cases of Practical Application

Case 1: Investment Decision

Suppose there are two investment alternatives:

• Alternative A: a gain of 1000 yuan with probability 60%, a loss of 500 yuan with probability 40%
• Alternative B: a gain of 300 yuan with probability 80%, a loss of 200 yuan with probability 20%

Compute the expected return of each alternative:

Expected return of alternative A: $E(A) = 0.6 \times 1000 + 0.4 \times (-500) = 600 - 200 = 400$

Expected return of alternative B: $E(B) = 0.8 \times 300 + 0.2 \times (-200) = 240 - 40 = 200$

Therefore, alternative A has the higher expected return.

Case 2: Quality Control

On a production line, the pass rate of products is 95%. For every 100 products produced, the expected number of qualified products is: $E(X) = 100 \times 0.95 = 95$

Case 3: Insurance Pricing

Insurance companies compute the expected loss of a certain class of accidents from historical data and set premiums on that basis.

Exercises

Exercise 1

Given that $X, Y$ are independent with $E(X)=1, E(Y)=2$, find $E(X+Y)$.

Exercise 2

A store sells a certain product. The profit per item is 10 yuan, and the distribution of the number of items sold is:

• Probability of selling 1 item: 0.3
• Probability of selling 2 items: 0.5
• Probability of selling 3 items: 0.2

Find the expected profit.

Exercise 3

The pass rate of the products of a factory is 90%. Unqualified products need to be reworked, at a rework cost of 50 yuan. The production cost of each item is 100 yuan. Find the expected cost per item.