📂Mathematical Statistics

Convolution Formula of Probability Density Functions

Formula ¹

Given two independent continuous random variables $X, Y$, their probability density functions are given by $f_{X}, f_{Y}$. Then the probability density function of $Z := X + Y$ is the convolution of the two probability density functions $f_{Z} = f_{X} \ast f_{Y}$. $$ f_{Z} (z) = \left( f_{X} \ast f_{Y} \right) (z) = \int_{-\infty}^{\infty} f_{X} (w) f_{Y} (z-w) dw $$

Derivation

If we let $W := X$, the Jacobian is $$ \begin{Vmatrix} 1 & 1 \\ 1 & 0 \end{Vmatrix} = \left| -1 \right| = 1 $$, and the joint probability density function of $Z$ and $W$ $f_{Z,W}$ is $$ f_{Z,W} \left( z,w \right) = f_{X,Y} \left( w, z-w \right) = f_{X} (w) f_{Y} (z-w) $$. Therefore, the marginal probability density function of $Z$ is found through the definite integral at $-\infty < w < \infty$ as follows. $$ f_{Z} (z) = \int_{-\infty}^{\infty} f_{X} (w) f_{Y} (z-w) |1| dw $$

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