1.	An investment will be worth $1,000, $2,000, or $5,000 at the end of the year. The probabilities of these values are .25, .60, and .15, respectively. Determine the mean and variance of the worth of the investment. 

To determine the **mean** and **variance** of the worth of the investment, we'll use the following formulas:

1. **Mean (Expected Value)**:  
\[
\mu = E(X) = \sum_{i=1}^{n} x_i \cdot p_i
\]
Where $x_i$ are the possible outcomes (investment values), and $p_i$ are the associated probabilities.

2. **Variance**:
\[
\sigma^2 = Var(X) = \sum_{i=1}^{n} p_i \cdot (x_i - \mu)^2
\]
Where $\mu$ is the mean (calculated from the first step).

### Step 1: Calculate the Mean
The possible outcomes are:  
- $x_1 = 1000$, with probability $p_1 = 0.25$  
- $x_2 = 2000$, with probability $p_2 = 0.60$  
- $x_3 = 5000$, with probability $p_3 = 0.15$

Using the formula for the mean:
\[
\mu = (1000 \times 0.25) + (2000 \times 0.60) + (5000 \times 0.15)
\]
\[
\mu = 250 + 1200 + 750 = 2200
\]

So, the **mean** (expected value) is $\mu = 2200$.

### Step 2: Calculate the Variance
First, calculate the squared deviations from the mean for each outcome:
- For $x_1 = 1000$, $(x_1 - \mu)^2 = (1000 - 2200)^2 = 1440000$
- For $x_2 = 2000$, $(x_2 - \mu)^2 = (2000 - 2200)^2 = 40000$
- For $x_3 = 5000$, $(x_3 - \mu)^2 = (5000 - 2200)^2 = 7840000$

Now, compute the variance:
\[
\sigma^2 = (0.25 \times 1440000) + (0.60 \times 40000) + (0.15 \times 7840000)
\]
\[
\sigma^2 = 360000 + 24000 + 1176000 = 1560000
\]

Thus, the **variance** is $\sigma^2 = 1,560,000$.

### Final Answers:
- **Mean (Expected Value)**: $\mu = 2200$
- **Variance**: $\sigma^2 = 1,560,000$

Let me know if you'd like more details or further clarifications!

---

Here are 5 related questions:
1. How is standard deviation related to variance?
2. What is the expected value in a different scenario with other probabilities?
3. How can you apply these calculations to more complex investments?
4. What are the practical implications of a higher or lower variance in investments?
5. How would the mean change if the probabilities were adjusted?

**Tip**: The square root of the variance gives the standard deviation, which helps in understanding the dispersion of data.

An investment will be worth $1,000, $2,000, or $5,000 at the end of the year. The probabilities of these values are .25, .60, and .15, respectively. Determine the mean and variance of the worth of the investment.

Learn how to calculate the mean (expected value) and variance for an investment scenario where the potential values are $1,000, $2,000, or $5,000 with respective probabilities of 0.25, 0.60, and 0.15. Understand the core statistical concepts of expected value and variance in this detailed guide.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation