Sums of Continuous Random Variables

Definition: Convolution of two densitites:

$$(f*g)(z)= \int_{- \infty}^{\infty}f(z-y)g(y)dy= \int_{- \infty}^{\infty}f(x)g(z-x)dx$$

Sums:For X and Y two random variables, and Z their sum, the density of Z is

$$f_Z(z) = \int_{- \infty}^{+ \infty} \int_{- \infty}^{z-x} f(x, y) dydx$$

Now if the random variables are independent, the density of their sum is the convolution of their densitites.

Examples:

1.

Sum of two independent uniform random variables:

$$f_Z(z)=\int f_X(z-y)f_Y(y)dy$$

Now f_Y(y)=1 only in [0,1]

$$f_Z(z)=\int_0^1 f_X(z-y)dy$$

This is zero unless $0\leq z-y \leq 1$ ( $z-1 \leq y \leq z$), otherwise it is zero: Case 1: $0\leq z \leq 1$ $f_Z(z)=\int_0^z dy=z$
Case 2: $1 < z \leq 2$, we have $f_Z(z)=\int_{z-1}^1 dy= 2-z$ For z smaller than 0 or bigger than 2 the density is zero. This density is triangular.

2.

Density of two indendent exponentials with parameter $\lambda$. $f_Z(z)=\lambda^2 z e^{-\lambda z}$, for z>0