91Ó°ÊÓ

a.

The cumulative distribution function of a normal random variable with mean $0$and variance $1$is localid="1649443303171" $\mathrm{Ï•}(-\mathrm{x})$. (that is, the standard normal random variable).

Because $\mathrm{Ï•}\left(\mathrm{x}\right)$is not symmetric about $0$, localid="1649443289397" $\mathrm{Ï•}(-\mathrm{x})=\mathrm{Ï•}\left(\mathrm{x}\right)$is incorrect (as the probability density function of the normal random variable is symmetric about $0$instead of the cumulative distribution function).

The equation $\mathrm{Ï•}(-\mathrm{x})=1-\mathrm{Ï•}\left(\mathrm{x}\right)$is correct in general, while $\mathrm{Ï•}(-\mathrm{x})=\mathrm{Ï•}\left(\mathrm{x}\right)$is only correct when $\mathrm{x}=0$(which also corresponds to $\mathrm{Ï•}=0.5$).

b.

$\mathrm{Ï•}\left(\mathrm{x}\right)+\mathrm{Ï•}(-\mathrm{x})=1$Because the equation $\mathrm{Ï•}(-\mathrm{x})=1-\mathrm{Ï•}\left(\mathrm{x}\right)$holds for the standard normal random variable, is correct.

We also notice that $\mathrm{Ï•}\left(\mathrm{x}\right)+\mathrm{Ï•}(-\mathrm{x})=1$resulted by adding $\mathrm{Ï•}\left(\mathrm{x}\right)$ in equation$\mathrm{Ï•}(-\mathrm{x})=1-\mathrm{Ï•}\left(\mathrm{x}\right)$

c.

Because $\mathrm{Ï•}(-\mathrm{x})$and $\mathrm{Ï•}\left(\mathrm{x}\right)$both represent probabilities, the equation$\mathrm{Ï•}(-\mathrm{x})=\frac{1}{\mathrm{Ï•}\left(\mathrm{x}\right)}$is incorrect.

This means that $\mathrm{Ï•}(-\mathrm{x})$and $\mathrm{Ï•}\left(\mathrm{x}\right)$are both values between $0$and$1$ (inclusive).