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MA181 INTRODUCTION TO STATISTICAL MODELLING

NORMAL DISTRIBUTION

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DIAGRAMS

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\item[Origins]
The normal distribution was discovered, in a discrete form, by de
Moivre in 1733 as an approximation to the binomial distribution.
It was later shown, in 1812, to be the limiting distribution of a
sample mean by Laplace. Meanwhile, in 1809, Gauss derived the
normal as the distribution of errors in astronomical observations.

\item[Formulation]
Let $Y$ be a random variable with the probability density function
(pdf)

$$f(y)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{(y-\mu)^2}{2\sigma^2}\right],\
-\infty<y<\infty.$$

Then $Y$ is said to follow a \textit{normal} (or
\textit{Gaussian})\textit{distribution} with parameters $\mu$ and
$\sigma^2$, the shorthand for which is $Y\sim N(\mu,\sigma^2)$.
The cumulative distribution function (cdf) of $Y$ is

$$F(y)=P(Y\leq
y=\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^y\exp\left[-\frac{(w-\mu)^2}{2\sigma^2}\right]\,dw,\
-\infty<y<\infty.$$

Clearly, $f(-\infty)=0$, and it can be shown that $F(\infty)=1$.

\item[Moments]
The moment generating function of $Y$ is given by

\begin{eqnarray*}
M(t)&=&\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^\infty
e^{ty}\exp\left[-\frac{(y-\mu)^2}{2\sigma^2}\right]\,dy\\
&=&e^{ut+\frac{1}{2}\sigma^2t^2}\times\frac{1}{\sigma\sqrt{2\pi}}
\int_{-\infty}^\infty\exp\left\{-\frac{[y-(\mu+\sigma^2t)]^2}{2\sigma^2}\right\}\,dy\\
&=&e^{\mu t+\frac{1}{2}\sigma^2t^2}\\ &=&1+(\mu
t+\frac{1}{2}\sigma^2t^2)+\frac{1}{2}(\mu
t+\frac{1}{2}\sigma^2t^2)^2+\ldots
\end{eqnarray*}

Consequently, $E(Y)=\mu$ and $E(Y^2)=\sigma^2+\mu^2$ so that
var$(Y)=\sigma^2$ and $\sigma$ is the standard deviation of $Y$.

As the normal distribution is symmetric $[f(\mu-c)=f(\mu+c)$ for
all $c]$, all its odd central moments are zero. On the other hand,
$\mu_4=E(Y-\mu)^4]=3\sigma^4$.

\item[Probabilities]
Whatever the values of $\mu$ and $\sigma$, the following
properties hold:

$\mu\mp\sigma$ contains 68.27\% of the distribution,
$\mu\mp1.960\sigma$ contains 95\%, $\mu\mp2.576\sigma\ 99\%$ and
$\mu\,mp3.291\sigma\ 99.9\%$.

\item[Standardisation]
If $Y'=a+bY$, then $Y'\sim N(a+b\mu,b^2\sigma^2)$. Hence. if
$Z=(Y-\mu)/\sigma$, then $Z\sim N(0,1)$, which is known as the
\textit{standard normal distribution} and is the only one
tabulated. The pdf of $Z$ is often denoted by $\phi(x)$ and the
cdf by $\phi(z)$.

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