Exercise 5.1.13

Answers

\begin{array}{l} 2 π \int_{\infty}^{\infty} p (x) dx \int_{\infty}^{\infty} p (y) dy & = 2 π \int_{\infty}^{\infty} \int_{\infty}^{\infty} p (x) p (y) dxdy \\ = 2 π \int_{\infty}^{\infty} \int_{\infty}^{\infty} \frac{1}{\sqrt{2 π}} e^{- \frac{x^{2}}{2}} \frac{1}{\sqrt{2 π}} e^{- \frac{y^{2}}{2}} dxdy \\ = \int_{\infty}^{\infty} \int_{\infty}^{\infty} e^{- \frac{x^{2} + y^{2}}{2}} dxdy \\ = \int_{0}^{2 π} \int_{0}^{\infty} e^{- \frac{r^{2}}{2}} rdrd𝜃 \\ = - \int_{0}^{2 π} e^{- \frac{r^{2}}{2}} |_{0}^{\infty} d𝜃 \\ = 2 π \end{array}

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2020-03-20 00:00