From this question here, I learned that the Cumulative Distribution Function (CDF) of $Z=X+Y$ is:
\begin{eqnarray*}
F_Z \left( z \right) & = & \int F_X \left( z – y
\right) dF_Y \left( y \right)\\
& = & \int F_Y \left( z – x \right) dF_X \left( x
\right)
\end{eqnarray*}
And Probability Density Function (PDF) of $Z=X+Y$ is:
\begin{eqnarray*}
f_Z \left( z \right) & = & \int f_X \left( z – y
\right) f_Y \left( y \right)dy\\
& = & \int f_Y \left( z – x \right) f_X \left( x
\right)dx
\end{eqnarray*}
How about PDF and CDF of sum of weighted independent random variables $Z=aX+bY$ where $a$ and $b$ are positive integers? I tried finding in this site but no luck. Thanks
Best Answer
\begin{align*} F_Z(z)&=Pr(aX+bY<z)=\int Pr(aX+by<z)f_Y(y)dy\\ &=\int F_X\left(\frac{z-by}{a}\right)f_Y(y)dy\\ f_Z(z)&=\frac{1}{|a|}\int f_X\left(\frac{z-by}{a}\right)f_Y(y)dy \end{align*}