Let $X_1$ and $X_2$ be two independent random variables (R.V.s), where $X_1$ follows a Poisson distribution with parameter $\lambda_1$ and $X_2$ follows a Poisson distribution with parameter $\lambda_2$. That is:

$X_1 \sim \text{Poisson}(\lambda_1)$ $X_2 \sim \text{Poisson}(\lambda_2)$

The probability mass function (PMF) of a Poisson-distributed random variable $X$ with parameter $\lambda$ is given by:

$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots$

Now, let $Y = X_1 + X_2$. We need to find the probability distribution function of $Y$.

Step 1: Determine the PMF of $Y$.

Since $X_1$ and $X_2$ are independent, the probability generating function of $Y = X_1 + X_2$ can be obtained by the product of the generating functions of $X_1$ and $X_2$.

The probability generating function of a Poisson random variable $X$ with parameter $\lambda$ is given by:

$G_X(t) = \mathbb{E}(t^X) = e^{\lambda(t-1)}$

Thus, the generating function of $Y$ is:

$G_Y(t) = G_{X_1}(t) \cdot G_{X_2}(t) = e^{\lambda_1(t-1)} \cdot e^{\lambda_2(t-1)} = e^{(\lambda_1 + \lambda_2)(t-1)}$

Step 2: Identify the Distribution of $Y$.

The generating function $G_Y(t) = e^{(\lambda_1 + \lambda_2)(t-1)}$ corresponds to a Poisson distribution with parameter $\lambda_1 + \lambda_2$.

Therefore, $Y$ follows a Poisson distribution with parameter $\lambda_1 + \lambda_2$. This means:

$Y \sim \text{Poisson}(\lambda_1 + \lambda_2)$

Conclusion:

The probability distribution function of $Y = X_1 + X_2$ is:

$P(Y = k) = \frac{(\lambda_1 + \lambda_2)^k e^{-(\lambda_1 + \lambda_2)}}{k!}, \quad k = 0, 1, 2, \dots$

Would you like more details or have any questions?

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Relative Questions:

1. How does the independence of $X_1$ and $X_2$ affect the distribution of $Y$?
2. What would happen if $X_1$ and $X_2$ were not independent?
3. How can the result be generalized for the sum of more than two independent Poisson random variables?
4. Can the sum of two Poisson-distributed random variables ever be non-Poisson?
5. How does the convolution of probability distributions relate to this problem?

Tip: When dealing with sums of independent random variables, consider using generating functions or moment-generating functions to identify the resulting distribution.