Question

If $X$ and $Y$ are independent Poisson variates with means as 10 and 5 respectively. Then $V(X - 2Y) =$

• A. 10
• B. 5
• C. 30
• D. 20

Hint:

Because variance measures spread, scaling constants are always squared when pulled out of a variance operator: $\text{Var}(-2Y) = (-2)^2\text{Var}(Y) = 4\text{Var}(Y)$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For independent random variables, the variance of a linear combination is computed by summing the scaled variances of the individual variables.
Key Formula or Approach:
1. For any Poisson random variable with mean rate $\lambda$, the variance is equal to its mean:
\[ \text{Var}(X) = E(X) = \lambda \]
2. For independent random variables $X$ and $Y$ and constants $a$ and $b$:
\[ \text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) \]

Step 2: Detailed Explanation:
We are given:
$X$ is a Poisson variate with mean 10 $\implies \text{Var}(X) = 10$
$Y$ is a Poisson variate with mean 5 $\implies \text{Var}(Y) = 5$
We need to find $V(X - 2Y)$.
Apply the variance properties for independent variables:
\[ \text{Var}(X - 2Y) = \text{Var}(X) + (-2)^2 \text{Var}(Y) \]
\[ \text{Var}(X - 2Y) = \text{Var}(X) + 4 \text{Var}(Y) \]
Substitute the known variances into the equation:
\[ \text{Var}(X - 2Y) = 10 + 4 \cdot (5) \]
\[ \text{Var}(X - 2Y) = 10 + 20 = 30 \]
Therefore, the variance of $X - 2Y$ is 30.

Step 3: Final Answer
The correct option is (C).