Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability density function ▷ Probability density function of sum of independents

Theorem: Let $X$ and $Y$ be two independent continuous random variables with possible values $\mathcal{X}$ and $\mathcal{Y}$ and let $Z = X + Y$. Then, the probability density function of $Z$ is given by

\[\label{eq:pdf-sumind} \begin{split} f_Z(z) &= \int_{-\infty}^{+\infty} f_X(z-y) f_Y(y) \, \mathrm{d}y \\ \text{or} \quad f_Z(z) &= \int_{-\infty}^{+\infty} f_Y(z-x) f_X(x) \, \mathrm{d}x \end{split}\]

where $f_X(x)$, $f_Y(y)$ and $f_Z(z)$ are the probability density functions of $X$, $Y$ and $Z$.

Proof: The cumulative distribution function of a sum of independent random variables is

\[\label{eq:cdf-sumind} F_Z(z) = \mathrm{E}\left[ F_X(z-Y) \right] \; .\]

The probability density function is the first derivative of the cumulative distribution function, such that

\[\label{eq:pdf-sumind-qed} \begin{split} f_Z(z) &= \frac{\mathrm{d}}{\mathrm{d}z} F_Z(z) \\ &= \frac{\mathrm{d}}{\mathrm{d}z} \mathrm{E}\left[ F_X(z-Y) \right] \\ &= \mathrm{E}\left[ \frac{\mathrm{d}}{\mathrm{d}z} F_X(z-Y) \right] \\ &= \mathrm{E}\left[ f_X(z-Y) \right] \\ &= \int_{-\infty}^{+\infty} f_X(z-y) f_Y(y) \, \mathrm{d}y \; . \end{split}\]

The second equation can be derived by switching $X$ and $Y$.

∎
Sources:

Metadata: ID: P258 | shortcut: pdf-sumind | author: JoramSoch | date: 2021-08-30, 09:31.