Sum of Random Variable
Overview
{% Z = X + Y %}
Discrete Probability
{% p_z(z) = \sum_{- \infty} ^{\infty} p(x, z-x) %}
Continuous Probability
Given the probability density functions for x and y, the cumulative function for z can be computed.
{% F_z(z) = \int_{- \infty} ^{\infty} \int_{- \infty} ^{z} f(x, v-x) dv dx %}
Interchanging the order of integration
{% F_z(z) = \int_{- \infty} ^{z} \int_{- \infty} ^{\infty} f(x, v-x) dx dv %}
Then the density function for z can be computed by taking the derivative
{% f_z(z) = \int_{- \infty} ^{\infty} f(x, z-x) dx %}
