Sum of independent variables

Definition:

• Sum of multiple Independent Random Variable, then they are Jointly distributed random variable
• Suppose that $X$ and $Y$ are independent, continuous random variables having PDF $f_X$ and $f_Y$.
• Then the CDF of $X+Y$ is: $F_{X+Y}=P(X+Y<a)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{a-y}{f}_X(x).f_Y(y)dxdy=\{\begin{array}{l}{\int }_{-\infty }^{\infty }{F}_X(a-y)f_Y(y)dy\\ {\int }_{-\infty }^{\infty }{F}_Y(a-x)f_X(x)dx\end{array}$
  • Also called the convolution of the distributions $F_X$ and $F_Y$
  • $f_{X+Y}={\int }_{-\infty }^{\infty }{f}_X(a-y)f_Y(y)dy={\int }_{-\infty }^{\infty }{f}_Y(a-x)f_X(x)dx$