Chapter 15.8, Problem 1P

Verify that for a random variable $x$ with normal density function $f(x)$ as in (8.1) the mean value of $x$ is $\mu ,$ the standard deviation is $\sigma ,$ and the integral of $f(x)$ from $-\infty$ to $\infty$ is 1 as it must be for a probability function. Hint: Write and evaluate the integrals ${\displaystyle {\int }_{-\infty }^{\infty }f}(x)dx,{\displaystyle {\int }_{-\infty }^{\infty }x}f(x)dx,{\displaystyle {\int }_{-\infty }^{\infty }{(x-\mu )}^2}f(x)dx.$ See equations (6.2) $(6.3),$ and (6.4).