Suppose X and Y are random variables with E(X) =10, std dev (X) = 3, E(Y) =15, and std dev (Y)= 4. Given that X and Y are independent, what are the mean and standard deviation of the random variable X - Y?

**Explain exactly how you calculated your final answer.**

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.

Suppose X and Y are random variables with E(X) = 10, std dev (X) = 3, E(Y) = 15, and std dev (Y) = 4. Given that X and Y are independent, what are the mean and standard deviation of the random variable X - Y?

Learn how to calculate the mean and standard deviation of the difference between two independent random variables X and Y, given E(X) = 10, std dev(X) = 3, E(Y) = 15, and std dev(Y) = 4. This solution uses the linearity of expectation and variance addition for independent variables.

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