Concept

Intuition through Convolutions

[Chan, 2021] section 5.5.1.

Convolutions of Random Variables

Theorem 44 (Convolutions of Random Variables)

Let \(X\) and \(Y\) be two independent random variables with PDFs \(f_X\) and \(f_Y\), respectively. Define \(Z = X + Y\) where \(Z\) is in itself a random variable. Then, the PDF of \(Z\) is given by

\[ f_Z(z) = \left(f_X \ast f_Y\right)(z) = \int_{-\infty}^{\infty} f_X(x) f_Y(z - x) \, \mathrm{d}x \]

where \(\ast\) denotes convolution.

Sum of Common Distribution

The following proofs are from [Chan, 2021] section 5.5.3. Sum of common distributions.

Theorem 45 (Sum of Poisson Random Variables)

Let \(X_1 \sim \operatorname{Poisson}\left(\lambda_1\right)\) and \(X_2 \sim \operatorname{Poisson}\left(\lambda_2\right)\). Then

\[ X_1+X_2 \sim \operatorname{Poisson}\left(\lambda_1+\lambda_2\right) \]

Proof. Let us apply the convolution principle.

\[\begin{split} \begin{aligned} & p_Y(k)=\mathbb{P}\left[X_1+X_2=k\right] \\ & =\mathbb{P}\left[X_1=\ell \cap X_2=k-\ell\right] \\ & =\sum_{\ell=0}^k \frac{\lambda_1^{\ell} e^{-\lambda_1}}{\ell !} \cdot \frac{\lambda_2^{k-\ell} e^{-\lambda_2}}{(k-\ell) !} \\ & =e^{-\left(\lambda_1+\lambda_2\right)} \sum_{\ell=0}^k \frac{\lambda_1^{\ell}}{\ell !} \cdot \frac{\lambda_2^{k-\ell}}{(k-\ell) !} \\ & =e^{-\left(\lambda_1+\lambda_2\right)} \cdot \frac{1}{k !} \underbrace{\sum_{\ell=0}^k \frac{k !}{\ell !(k-\ell) !} \lambda_1^{\ell} \lambda_2^{k-\ell}}_{=\sum_{\ell=0}^k\left(\begin{array}{l} k \\ \ell \end{array}\right) \lambda_1^{\ell} \lambda_2^{k-\ell}} \\ & =\frac{\left(\lambda_1+\lambda_2\right)^k}{k !} e^{-\left(\lambda_1+\lambda_2\right)} \text {, } \\ & \end{aligned} \end{split}\]

where the last step is based on the binomial identity \(\sum_{\ell=0}^k\left(\begin{array}{c}k \\ \ell\end{array}\right) a^{\ell} b^{k-\ell}=(a+b)^k\).

Theorem 46 (Sum of Gaussian Random Variables)

Let \(X_1\) and \(X_2\) be two Gaussian random variables such that

\[ X_1 \sim \operatorname{Gaussian}\left(\mu_1, \sigma_1^2\right) \quad \text { and } \quad X_2 \sim \operatorname{Gaussian}\left(\mu_2, \sigma_2^2\right) . \]

Then

\[ X_1+X_2 \sim \operatorname{Gaussian}\left(\mu_1+\mu_2, \sigma_1^2+\sigma_2^2\right) . \]

Proof. Let us apply the convolution principle.

\[\begin{split} \begin{aligned} f_Z(z) & =\int_{-\infty}^{\infty} f_X(t) f_Y(z-t) d t \\ & =\int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left\{-\frac{\left(t-\mu_1\right)^2}{2 \sigma^2}\right\} \cdot \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left\{-\frac{\left(z-t-\mu_2\right)^2}{2 \sigma^2}\right\} d t \\ & =\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left\{-\frac{\left(t-\mu_1\right)^2+\left(z-t-\mu_2\right)^2}{2 \sigma^2}\right\} d t . \end{aligned} \end{split}\]

We now complete the square:

\[\begin{split} \begin{aligned} \left(t-\mu_1\right)^2+(z & \left.-t-\mu_2\right)^2=\left[t^2-2 \mu_1 t+\mu_1^2\right]+\left[t^2+2 t\left(\mu_2-z\right)+\left(\mu_2-z\right)^2\right] \\ & =2 t^2-2 t\left(\mu_1-\mu_2+z\right)+\mu_1^2+\left(\mu_2-z\right)^2 \\ & =2\left[t^2-2 t \cdot \frac{\mu_1-\mu_2+z}{2}\right]+\mu_1^2+\left(\mu_2-z\right)^2 \\ & =2\left[t-\frac{\mu_1-\mu_2+z}{2}\right]^2-2\left[\frac{\mu_1-\mu_2+z}{2}\right]^2+\mu_1^2+\left(\mu_2-z\right)^2 \end{aligned} \end{split}\]

The last term can be simplified to

\[\begin{split} \begin{aligned} -2 & {\left[\frac{\mu_1-\mu_2+z}{2}\right]^2+\mu_1^2+\left(\mu_2-z\right)^2 } \\ & =-\frac{\mu_1^2-2 \mu_1\left(\mu_2-z\right)+\left(\mu_2-z\right)^2}{2}+\mu_1^2+\left(\mu_2-z\right)^2 \\ & =\frac{\mu_1^2+2 \mu_1\left(\mu_2-z\right)+\left(\mu_2-z\right)^2}{2}=\frac{\left(\mu_1+\mu_2-z\right)^2}{2} . \end{aligned} \end{split}\]

Substituting these into the integral, we can show that

\[\begin{split} \begin{aligned} f_Z(z) & =\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left\{-\frac{2\left[t-\frac{\mu_1-\mu_2+z}{2}\right]^2+\frac{\left(\mu_1+\mu_2-z\right)^2}{2}}{2 \sigma^2}\right\} d t \\ & =\frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left\{-\frac{\left(\mu_1+\mu_2-z\right)^2}{2\left(2 \sigma^2\right)}\right\} \underbrace{\int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left\{-\frac{\left[t-\frac{\mu_1-\mu_2+z}{2}\right]^2}{\sigma^2}\right\} d t}_{=\frac{1}{\sqrt{2}}} \\ & =\frac{1}{\sqrt{2 \pi(2 \sigma)^2}} \exp \left\{-\frac{\left(\mu_1+\mu_2-z\right)^2}{2\left(2 \sigma^2\right)}\right\} . \end{aligned} \end{split}\]

Therefore, we have shown that the resulting distribution is a Gaussian with mean \(\mu_1+\mu_2\) and variance \(2 \sigma^2\).

Theorem 47 (Sum of Common Distributions)

Let \(X_1\) and \(X_2\) be two independent random variables that come from the same family of distributions.

Then, the PDF of \(X_1 + X_2\) is given by

Table 15 Sum of Common Distributions

\(X_1\)	\(X_2\)	\(X_1 + X_2\)
\(\bern(p)\)	\(\bern(p)\)	\(\binomial(n, p)\)
\(\binomial(n, p)\)	\(\binomial(m, p)\)	\(\binomial(m+n, p)\)
\(\poisson(\lambda_1)\)	\(\poisson(\lambda_2)\)	\(\poisson(\lambda_1 + \lambda_2)\)
\(\exponential(\lambda)\)	\(\exponential(\lambda)\)	\(\operatorname{Erlang}(2, \lambda)\)
\(\gaussian(\mu_1, \sigma_1)\)	\(\gaussian(\mu_2, \sigma_2)\)	\(\gaussian(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2)\)

This holds for \(N\) random variables as well.