Provide an Appropriate Response $f ( x ) = \frac { 1 } { \sqrt { 2 \pi } } e ^ { - x ^ { 2 } / 2 }$

Essay

Provide an appropriate response.
-The standard normal probability density function is defined by $f ( x ) = \frac { 1 } { \sqrt { 2 \pi } } e ^ { - x ^ { 2 } / 2 }$ .
(a) Use the fact that $\int _ { - \infty } ^ { \infty } \frac { 1 } { \sqrt { 2 \pi } } e ^ { - } x ^ { 2 } / 2 d x = 1$ to show that $\int _ { 0 } ^ { \infty } \frac { 1 } { \sqrt { 2 \pi } } x ^ { 2 } e ^ { - x ^ { 2 } / 2 } \mathrm { dx } = \frac { 1 } { 2 }$ .
(b) Use the result in part (a) to show that the standard normal probability density function has variance $1 .$