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Let's break down the problem step-by-step.

### Problem Summary:
- **Person 1:**
  - Expected value $ E_1 = 31,000 $
  - Standard deviation $ \sigma_1 = 3,000 $
- **Person 2:**
  - Expected value $ E_2 = 33,000 $
  - Standard deviation $ \sigma_2 = 5,000 $

We need to determine the following for the total earnings of the family:

1. **Expected Value** $ E $
2. **Standard Deviation** $ \sigma $

### (a) Expected Value
The expected value of the total earnings is the sum of the expected values of each person, because expectation is a linear operator.

\[
E = E_1 + E_2 = 31,000 + 33,000 = 64,000
\]

### (b) Standard Deviation
To find the standard deviation, we first need to calculate the variance of the total earnings. The variance of the sum of two independent random variables is the sum of their variances.

\[
\text{Variance} = \sigma_1^2 + \sigma_2^2
\]

Calculating:

\[
\text{Variance} = (3,000)^2 + (5,000)^2 = 9,000,000 + 25,000,000 = 34,000,000
\]

The standard deviation is the square root of the variance:

\[
\sigma = \sqrt{34,000,000} \approx 5,831.0
\]

### Summary
1. **Expected Value** of total earnings: $ 64,000 $
2. **Standard Deviation** of total earnings: $ 5,831.0 $

Would you like more details on any step or do you have any questions? Here are 5 related questions for further exploration:

1. How would the total expected value change if the second person's expected earnings increased by $5,000?
2. What would happen to the total standard deviation if both individuals' earnings had the same standard deviation?
3. How does the assumption of independence between earnings affect the calculation of variance?
4. What is the probability that the total earnings exceed $70,000, assuming normal distribution?
5. How would the standard deviation change if the first person's standard deviation doubled?

**Tip:** When dealing with independent random variables, remember that the variance of their sum is the sum of their variances, which simplifies calculations!

In a two-person family, the amount of money that the first person earns has expected value $31,000 and standard deviation $3,000. The second person earns an expected value of $33,000 with a standard deviation of $5,000. Determine the (a) expected value and (b) standard deviation of the total earnings of this family.

Learn how to calculate the expected value and standard deviation of a family's total earnings. This problem involves independent random variables, expected values, and variance calculations, and provides step-by-step guidance.

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