To find the mean and standard deviation of the random variable $X - Y$, we can use the properties of expectations and variances of independent random variables.

Step 1: Calculate the Mean of $X - Y$

The expectation of a difference $X - Y$ of two random variables $X$ and $Y$ can be calculated as: $E(X - Y) = E(X) - E(Y)$ Given that:

• $E(X) = 10$
• $E(Y) = 15$

We substitute these values: $E(X - Y) = 10 - 15 = -5$

So, the mean of $X - Y$ is $-5$.

Step 2: Calculate the Variance of $X - Y$

Since $X$ and $Y$ are independent, the variance of $X - Y$ is: $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$

To calculate this, we need the variances of $X$ and $Y$. Given the standard deviations:

• $\text{std dev}(X) = 3$
• $\text{std dev}(Y) = 4$

The variances are: $\text{Var}(X) = (\text{std dev}(X))^2 = 3^2 = 9$ $\text{Var}(Y) = (\text{std dev}(Y))^2 = 4^2 = 16$

Substitute these values into the variance formula: $\text{Var}(X - Y) = 9 + 16 = 25$

Step 3: Calculate the Standard Deviation of $X - Y$

The standard deviation of $X - Y$ is the square root of the variance: $\text{std dev}(X - Y) = \sqrt{\text{Var}(X - Y)} = \sqrt{25} = 5$

Final Answer

• The mean of $X - Y$ is $-5$.
• The standard deviation of $X - Y$ is $5$.

Summary of Calculations

1. Mean: $E(X - Y) = E(X) - E(Y) = 10 - 15 = -5$
2. Variance: $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) = 9 + 16 = 25$
3. Standard Deviation: $\text{std dev}(X - Y) = \sqrt{25} = 5$

Would you like further details, or do you have any questions?

Here are some additional related questions:

1. How would the calculations change if $X$ and $Y$ were not independent?
2. What is the variance of $X + Y$ given that $X$ and $Y$ are independent?
3. How would you calculate $E(X + Y)$ and $\text{std dev}(X + Y)$?
4. If $X$ and $Y$ had a correlation, how would that affect the variance of $X - Y$?
5. Can you calculate $E(aX + bY)$ and $\text{std dev}(aX + bY)$ for constants $a$ and $b$?

Tip: Remember that for independent variables, the variance of their sum or difference is the sum of their variances, regardless of the signs.