Problem Explanation:

Part 1:

The first part requires proving that if $X$ and $Y$ are independent random variables with probability density functions $p_X(x)$ and $p_Y(y)$, then the probability density function $p_Z(z)$ of their sum $Z = X + Y$ is given by: $p_Z(z) = \int_{\mathbb{R}^D} p_X(x) p_Y(z - x) dx$ This result comes from the convolution of the probability density functions of independent random variables.

Part 2:

The second part involves explicitly computing $p_Z(z)$ when $X \sim \mathcal{N}(\mu_X, \sigma_X^2)$ and $Y \sim \mathcal{N}(\mu_Y, \sigma_Y^2)$, which are one-dimensional independent normal random variables.

Solution:

1. Proof of Convolution Formula:

Let $Z = X + Y$. The probability density function of $Z$, $p_Z(z)$, is defined as: $p_Z(z) = P(Z = z) = P(X + Y = z).$ By the independence of $X$ and $Y$, their joint density is the product of their individual densities. So, we can write: $p_Z(z) = \int_{x \in \mathbb{R}^D} p_X(x) p_Y(z - x) dx.$ This is the convolution formula for the sum of two independent random variables.

2. Sum of Two Normal Distributions:

Let $X \sim \mathcal{N}(\mu_X, \sigma_X^2)$ and $Y \sim \mathcal{N}(\mu_Y, \sigma_Y^2)$.

2.1 Mean of $Z$:

The expectation of $Z$ is: [ \mathbb{E}[Z] = \mathbb{E}[X + Y] = \mathbb{E}[X] + \mathbb{E}[Y] = \mu_X + \mu_Y. ]

2.2 Variance of $Z$:

Since $X$ and $Y$ are independent, their variances add: $\text{Var}(Z) = \text{Var}(X) + \text{Var}(Y) = \sigma_X^2 + \sigma_Y^2.$

2.3 Distribution of $Z$:

The sum of two independent normal random variables is also normally distributed: $Z \sim \mathcal{N}(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2).$ Thus, the probability density function of $Z$ is: $p_Z(z) = \frac{1}{\sqrt{2\pi (\sigma_X^2 + \sigma_Y^2)}} \exp\left(-\frac{(z - (\mu_X + \mu_Y))^2}{2(\sigma_X^2 + \sigma_Y^2)}\right).$

Final Results:

1. Convolution Formula: $p_Z(z) = \int_{\mathbb{R}^D} p_X(x) p_Y(z - x) dx$

2. Probability Density Function for $Z$ (Normal Sum): $p_Z(z) = \frac{1}{\sqrt{2\pi (\sigma_X^2 + \sigma_Y^2)}} \exp\left(-\frac{(z - (\mu_X + \mu_Y))^2}{2(\sigma_X^2 + \sigma_Y^2)}\right).$

Let me know if you’d like a more detailed step-by-step derivation for either part!

Questions to Explore:

1. What happens to the convolution formula if $X$ and $Y$ are not independent?
2. How would the formula generalize to three independent random variables $X + Y + W$?
3. Can you derive the characteristic function approach to sum two independent normal variables?
4. What is the convolution formula in higher dimensions ($D > 1$)?
5. How can the convolution theorem be used in Fourier transforms for finding $p_Z(z)$?

Tip:

For sums of independent variables, the characteristic function (Fourier transform of the PDF) simplifies convolutions as it converts them into products.