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:

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.

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 $a^2$.