*X* and Y*Y* are independent random variables with means 10 and 17, and variances 5 and 5 respectively. Find the variance of 6X+3Y6*X*+3*Y*

Given that \( X \) and \( Y \) are independent random variables with:

- \( \text{E}(X) = 10 \)
- \( \text{E}(Y) = 17 \)
- \( \text{Var}(X) = 5 \)
- \( \text{Var}(Y) = 5 \)

We are asked to find the variance of the expression \( 6X + 3Y \).

### Step 1: Understanding the Variance of a Linear Combination

The variance of a linear combination of random variables is given by:
\[
\text{Var}(aX + bY) = a^2 \cdot \text{Var}(X) + b^2 \cdot \text{Var}(Y)
\]
where \( a \) and \( b \) are constants, and \( X \) and \( Y \) are independent random variables.

### Step 2: Apply the Formula

For the expression \( 6X + 3Y \):

- \( a = 6 \)
- \( b = 3 \)
  
Substitute the values into the formula:
\[
\text{Var}(6X + 3Y) = 6^2 \cdot \text{Var}(X) + 3^2 \cdot \text{Var}(Y)
\]
\[
\text{Var}(6X + 3Y) = 36 \cdot 5 + 9 \cdot 5
\]

### Step 3: Calculate the Variance

\[
\text{Var}(6X + 3Y) = 180 + 45 = 225
\]

### Final Answer:
The variance of \( 6X + 3Y \) is \( \boxed{225} \).

Do you want more details or have any questions?

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Here are 5 related questions:
1. What is the expected value of \( 6X + 3Y \)?
2. How does the variance change if the coefficients of \( X \) and \( Y \) are different?
3. If \( X \) and \( Y \) were not independent, how would you find the variance of \( 6X + 3Y \)?
4. What if \( X \) and \( Y \) had different variances?
5. How would you calculate the standard deviation of \( 6X + 3Y \)?

**Tip:** Remember that the variance of a linear combination depends on the square of the coefficients multiplied by the variances of the variables involved.

Discover how to calculate the variance of the expression 6X + 3Y, where X and Y are independent random variables with specified means and variances. Learn the step-by-step method using the formula for the variance of a linear combination. Suitable for advanced undergraduate students and those interested in probability theory.

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