To solve this, let's consider each transformation separately. Here’s the approach for calculating the new mean and standard deviation for each transformed variable.

Given Information:

1. $X$ has a mean $\mu_X = 10$ and standard deviation $\sigma_X = 2$.
2. $Y$ has a mean $\mu_Y = 12$ and standard deviation $\sigma_Y = 3$.

Transformed Variables

1. For $X - 20$:

   • Mean: $\mathbb{E}(X - 20) = \mathbb{E}(X) - 20 = 10 - 20 = -10$.
   • Standard Deviation: $\text{Var}(X - 20) = \text{Var}(X) = \sigma_X^2 = 4$, so $\sigma = \sqrt{4} = 2$.

   Result: Mean = -10, Standard Deviation = 2.

2. For $0.5Y$:

   • Mean: $\mathbb{E}(0.5Y) = 0.5 \cdot \mathbb{E}(Y) = 0.5 \cdot 12 = 6$.
   • Standard Deviation: $\text{Var}(0.5Y) = (0.5)^2 \cdot \text{Var}(Y) = 0.25 \cdot 9 = 2.25$, so $\sigma = \sqrt{2.25} = 1.5$.

   Result: Mean = 6, Standard Deviation = 1.5.

3. For $X + Y$:

   • Mean: $\mathbb{E}(X + Y) = \mathbb{E}(X) + \mathbb{E}(Y) = 10 + 12 = 22$.
   • Standard Deviation: Since $X$ and $Y$ are independent, $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) = 4 + 9 = 13$, so $\sigma = \sqrt{13} \approx 3.61$.

   Result: Mean = 22, Standard Deviation ≈ 3.61.

4. For $X - Y$:

   • Mean: $\mathbb{E}(X - Y) = \mathbb{E}(X) - \mathbb{E}(Y) = 10 - 12 = -2$.
   • Standard Deviation: Since $X$ and $Y$ are independent, $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) = 4 + 9 = 13$, so $\sigma = \sqrt{13} \approx 3.61$.

   Result: Mean = -2, Standard Deviation ≈ 3.61.

5. For $Y_1 + Y_2$ (assuming $Y_1$ and $Y_2$ are independent copies of $Y$):

   • Mean: $\mathbb{E}(Y_1 + Y_2) = \mathbb{E}(Y) + \mathbb{E}(Y) = 12 + 12 = 24$.
   • Standard Deviation: Since $Y_1$ and $Y_2$ are independent, $\text{Var}(Y_1 + Y_2) = \text{Var}(Y) + \text{Var}(Y) = 9 + 9 = 18$, so $\sigma = \sqrt{18} \approx 4.24$.

   Result: Mean = 24, Standard Deviation ≈ 4.24.

Summary of Results

• $X - 20$: Mean = -10, Standard Deviation = 2
• $0.5Y$: Mean = 6, Standard Deviation = 1.5
• $X + Y$: Mean = 22, Standard Deviation ≈ 3.61
• $X - Y$: Mean = -2, Standard Deviation ≈ 3.61
• $Y_1 + Y_2$: Mean = 24, Standard Deviation ≈ 4.24

Would you like further details on any of these calculations or another related question?

Related Questions

1. How would you calculate the variance of $X - 3Y$?
2. What if $Y$ had a different distribution type; how would that impact these transformations?
3. How can you interpret the result for $X + Y$ if $X$ and $Y$ are not independent?
4. How would you handle calculating $X \cdot Y$'s mean and variance given independence?
5. What if $X$ and $Y$ have non-zero covariance; how would you find the variance of $X + Y$?

Tip: For independent variables, sums and differences have variances that add up, irrespective of the sign of the terms, making calculations simpler.