I want to show that $\int_{\mathbb{R}} p_s(x-y)p_t(y-z) = p_{t+s}(x-z) $ where $p_t(x) = \frac{1}{\sqrt{2 \pi t}} \exp(\frac{-x^2}{2t})$.
My attempt: \begin{eqnarray*} \int_{\mathbb{R}} p_s(x-y)p_t(y-z) & = \int_{\mathbb{R}} \frac{1}{\sqrt{2 \pi s}}\exp(\frac{-(x-y)^2}{2s}) \frac{1}{\sqrt{2 \pi t}}\exp(\frac{-(y-z)^2}{2t})\\ & = \int_{\mathbb{R}} \frac{1}{2 \pi\sqrt{st}} \exp(\frac{-(x-y)^2}{2s} - \frac{(y-z)^2}{2t}) \end{eqnarray*} After this I am trying to do completing square form but stuck there and not able to reach to any conclusion. Can anyone give some hints to prove this identity ?