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.