https://file.aiquickdraw.com/mathbot/file/172610124607388uq5lyq.jpg

Let's break down the problem and verify the calculations for the random variables \( X \), \( Y = X + X \), and \( Z = X - X \).

### Step 1: Mean, Variance, and Standard Deviation of \( X \)
Given the probability distribution:

| \( X \) | 1   | 2   | 3   | 8   | 10  | 13  | 18  |
|--------|-----|-----|-----|-----|-----|-----|-----|
| \( P(X) \) | 0.06 | 0.09 | 0.07 | 0.13 | 0.48 | 0.11 | 0.06 |

The mean \( \mu_X \), variance \( \sigma_X^2 \), and standard deviation \( \sigma_X \) are already calculated as:

- \( \mu_X = 8.8 \)
- \( \sigma_X^2 = 17.96 \)
- \( \sigma_X = 4.24 \)

These values are correct.

### Step 2: Mean, Variance, and Standard Deviation of \( Y = X + X \)
For the variable \( Y = X + X = 2X \), the following formulas apply:

- Mean: \( \mu_Y = 2\mu_X = 2 \times 8.8 = 17.6 \) (which is correct)
- Variance: \( \sigma_Y^2 = 2^2 \cdot \sigma_X^2 = 4 \times 17.96 = 71.84 \) (The image shows \( 71.44 \), which is incorrect.)
- Standard deviation: \( \sigma_Y = \sqrt{\sigma_Y^2} = \sqrt{71.84} = 8.475 \) (not 8.45)

### Step 3: Mean, Variance, and Standard Deviation of \( Z = X - X \)
For \( Z = X - X = 0 \):

- Mean: \( \mu_Z = \mu_X - \mu_X = 0 \)
- Variance: \( \sigma_Z^2 = 0 \)
- Standard deviation: \( \sigma_Z = 0 \)

These values are indeed correct.

### Corrections
- For \( Y = X + X \): the variance should be \( 71.84 \), not \( 71.44 \), and the standard deviation should be \( 8.475 \), not \( 8.45 \).

Let me know if you would like a more detailed explanation or have further questions.

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Here are some follow-up questions for practice:

1. How do you calculate the variance of a linear transformation of a random variable?
2. Why is the mean of \( Z = X - X \) equal to 0?
3. Can you explain why the standard deviation of \( Z = X - X \) is 0?
4. What is the effect of multiplying a random variable by a constant on its variance?
5. How would the calculations change if we considered a different random variable \( W = 3X \)?

**Tip**: For any linear transformation \( aX + b \), the variance is scaled by \( a^2 \).

Consider the discrete random variable X given in the table below. Calculate the mean, variance, and standard deviation of X. Then, find the mean, variance, and standard deviation of Y = X + X and Z = X - X.

Learn how to calculate the mean, variance, and standard deviation for a discrete random variable X. Additionally, explore how linear transformations affect these calculations for Y = X + X and Z = X - X. Includes step-by-step solutions and formulas.

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