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.