Sum of independent random variables

Let $𝑋 1, 𝑋 2 : 𝜉 \to ℝ$ be independently distributed real random variables. Then the distribution of $𝑌 = 𝑋 1 + 𝑋 2$ is given by the convolution of that of $𝑋 1$ with that of $𝑋 2$ , prob i.e. in the discrete case the probability mass function is

𝑝 𝑌 (𝑦) = \sum 𝑥 1 \in s u p p (𝑋 1) 𝑝 𝑋 2 (𝑦 - 𝑥 1) 𝑝 𝑋 1 (𝑥 1)

and in the continuous case the probability density function is

𝑓 𝑌 (𝑦) = \int \infty - \infty 𝑓 𝑋 2 (𝑦 - 𝑥 1) 𝑓 𝑋 1 (𝑥 1) 𝑑 𝑥 1