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\def\ctitle{Math 203: Principles of Statistics 1}
\def\ttitle{Fall 2024}
\def\stitle{Normal Distribution Tables}

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\begin{document}

<<funcs,echo=FALSE>>=
myrnd <- function(x,d) {formatC(x, format="f", digits=d)}
library(shape)
library(diagram)
@

\maketitle
This Normal table displays the standard Normal cdf
\[
\Phi(z) = P(Z \leq z) = \int_{-\infty}^z f(t) \ dt
\]  
where $t$ is the dummy/integrating variable, and $f(.)$ is the standard Normal pdf.

<<chunk8,fig.height=5, fig.width=8,echo=FALSE,size='small'>>=
Phi.mat<-round(pnorm(outer(seq(0.0,3.5,by=0.1),seq(0.0,0.09,by=0.01),'+')),4)
rownames(Phi.mat)<-myrnd(seq(0.0,3.5,by=0.1),1)
colnames(Phi.mat)<-myrnd(seq(0.0,0.09,by=0.01),2)
Phi.mat
@
Due to the symmetry of the standard Normal pdf, for $z < 0$
\[
\Phi(z) = P(Z \leq z) = P(Z > -z) = 1 - P(Z \leq -z)  = 1 -\Phi(-z).
\]

The textbook (p. 816) provides a slightly different version, tabulating
\[
P(0 < Z \leq z) =  \int_{0}^z f(t) \ dt 
\]
that is, the area under the pdf between zero and $z$. 
\begin{figure}[ht]
  \centering
  \includegraphics[scale=0.8]{TextbookNormal.jpg}
\end{figure}

The numbers in this table can be used in the same way as the original table; the only difference is that 0.5 has been subtracted from the values in the first table.   This follows as
\[
\int_{-\infty}^z f(t) \ dt = \int_{-\infty}^0 f(t) \ dt + \int_{0}^z f(t) \ dt = 0.5 + \int_{0}^z f(t) \ dt.
\]
In the course, we will be using the first table rather than the textbook table.

\end{document}