Consider the function

$$\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad x \in \mathbb{R}$$

Show that $\phi$ defines a probability density function. If a random variable $X$ has probability density function $\phi$, find the moment generating function of $X$, and find all moments $E\left[X^{k}\right]$, $k \in \mathbb{N}$.

Now define

$$r(x)=\frac{P(X>x)}{\phi(x)}$$

Show that for every $x>0$,

$$\frac{1}{x}-\frac{1}{x^{3}}<r(x)<\frac{1}{x}$$