If X and Y are independent Poisson random variables with expectations 5 and 3 respectively, then

find the variance of 2X + Y .

To find the variance of the linear combination \(2X + Y\), where \(X\) and \(Y\) are independent Poisson random variables, you can follow these steps:

### 1. **Recall the Properties of Variance:**
   - For a random variable \(X\) with variance \(\text{Var}(X)\), the variance of a scaled random variable \(aX\) is:
     \[
     \text{Var}(aX) = a^2 \cdot \text{Var}(X)
     \]
   - If \(X\) and \(Y\) are independent random variables, the variance of their sum is:
     \[
     \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)
     \]

### 2. **Find the Variances of \(X\) and \(Y\):**
   - For a Poisson random variable, the variance is equal to the expectation. 
     - So, \(\text{Var}(X) = \mathbb{E}[X] = 5\).
     - And \(\text{Var}(Y) = \mathbb{E}[Y] = 3\).

### 3. **Compute the Variance of \(2X + Y\):**
   - Using the scaling property of variance:
     \[
     \text{Var}(2X) = 2^2 \cdot \text{Var}(X) = 4 \cdot 5 = 20
     \]
   - Since \(X\) and \(Y\) are independent:
     \[
     \text{Var}(2X + Y) = \text{Var}(2X) + \text{Var}(Y) = 20 + 3 = 23
     \]

### **Final Answer:**

The variance of \(2X + Y\) is \( \boxed{23} \).

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Would you like more details on this, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. What is the expectation of \(2X + Y\) given the expectations of \(X\) and \(Y\)?
2. How does the covariance between \(X\) and \(Y\) affect the variance of their sum?
3. What is the moment-generating function for a linear combination of Poisson random variables?
4. How do you find the variance of \(X - Y\) if \(X\) and \(Y\) are not independent?
5. How do you calculate the probability \(P(2X + Y = k)\) for some integer \(k\)?

**Tip:** Remember that the variance of independent random variables adds up directly, but for dependent variables, you need to consider their covariance as well.

Learn how to find the variance of 2X + Y where X and Y are independent Poisson random variables with expectations 5 and 3, respectively. Detailed step-by-step solution provided.

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