For a standard normal random variable Z $$ N $(0,1),$ we denote its:

$$ CDF $\backslash$Phi $($z$)=$ P$($Z $<=$ z$)=$ R z

$\backslash$infty $\backslash$phi $($z$)$dz$$ at a point z in R$,$ where $\backslash$phi is the PDF

of the N $(0,1)$ distribution;

$$ Quantile $($inverse CDF$)\backslash$Phi $1($p$)$ at a point p in $(0,1).$

In other words, if $\backslash$Phi $($z$)=$ p then $\backslash$Phi $1($p$)=$ z$.$

For the purposes of this problem sheet, we use the standard normal tables to

evaluate the CDF and its inverse approximately as follows:

$$ To evaluate $\backslash$Phi $($z$)$ at a point z$,$ we find the tabulated input z$$ closest to z and

return $\backslash$Phi $($z$)$;

$$ To evaluate $\backslash$Phi $1($p$)$ at a point p$,$ we find the tabulated output p$=\backslash$Phi $($z$)$

closest to p and return z$.$

If a point falls halfway between two tabulated values, then we may interpolate to

increase accuracy. For example, the tables give $\backslash$Phi $(1\mathrm{.}64)=0\mathrm{.}9495$ and $\backslash$Phi $(1\mathrm{.}65)=$

$0\mathrm{.}9505$ but not z such that $\backslash$Phi $($z$)=0\mathrm{.}95,$ so we may interpolate and use $\backslash$Phi $(1\mathrm{.}645)=$

$0\mathrm{.}95$ as a better approximation.

You may wish to check your calculations using the pnorm $($CDF$)$ and qnorm func$-$

tions in R or the equivalent functions in another programme. For example, if

X $$ N $(1,32),$ then we compute P$($X $<=2)$ by executing pnorm$(2,-1,3)$ in

R$$s command window. To see the help page for these functions, execute $?$pnorm

or $?$qnorm in the command window.