This problem involves the derivation of the probability density function $p_Z(z)$ for the sum $Z = X + Y$, where $X$ and $Y$ are independent random variables.

Proof:

1. Definition of $Z = X + Y$: Let $Z = X + Y$, where $X$ and $Y$ are independent random variables. The independence implies that their joint probability density function can be expressed as the product of their individual densities: $p_{X,Y}(x, y) = p_X(x) p_Y(y).$

2. Expression for $p_Z(z)$: By definition, the probability density function $p_Z(z)$ of $Z$ is given by: $p_Z(z) = \int_{\mathbb{R}^D} p_{X,Y}(x, z - x) \, dx,$ where $z \in \mathbb{R}^D$, and the integral is taken over the entire domain $\mathbb{R}^D$. This integral accounts for all possible ways that $X$ and $Y$ can sum to $Z$.

3. Substitution for independence: Since $X$ and $Y$ are independent, substitute $p_{X,Y}(x, z - x) = p_X(x) p_Y(z - x)$ into the integral: $p_Z(z) = \int_{\mathbb{R}^D} p_X(x) p_Y(z - x) \, dx.$

4. Final Result: The derived formula for $p_Z(z)$ is: $p_Z(z) = \int_{\mathbb{R}^D} p_X(x) p_Y(z - x) \, dx,$ which holds for $z \in \mathbb{R}^D$. This is the convolution of the probability density functions $p_X$ and $p_Y$.

Explanation:

• The convolution formula arises because for each fixed value of $z$, $X$ and $Y$ can contribute to $Z$ in a variety of ways, and the integral accumulates these contributions over all $x \in \mathbb{R}^D$.
• The assumption of independence is crucial in allowing the joint density to factorize into the product of $p_X$ and $p_Y$.

Let me know if you'd like me to elaborate further or address related topics!

Here are some additional questions to expand your understanding:

1. What happens if $X$ and $Y$ are not independent? How would the formula for $p_Z(z)$ change?
2. Can this formula be generalized for discrete random variables?
3. How does the dimensionality $D$ of $\mathbb{R}^D$ affect the integral?
4. How does convolution in probability theory relate to convolution in signal processing?
5. Can you find an example where $X$ and $Y$ are Gaussian random variables, and compute $p_Z(z)$?

Tip: Convolution in probability corresponds to adding random variables. Understanding this principle can help you analyze a variety of stochastic systems.