# Probability Density Function of Convolution of Probability Distributions

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## Theorem

Let $X$ and $Y$ be continuous independent random variables.

Let $Z = X + Y$.

Let $f_X$ be a probability density function for $X$.

Let $f_Y$ be a probability density function for $Y$.

Then a probability density function for $Z$, $f_Z$, is given by:

- $\ds \map {f_Z} z = \int_{-\infty}^\infty \map {f_X} x \map {f_Y} {z - x} \rd x$

## Proof

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