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Rb: /* Calculating probabilities when X\sim N(\protect\mu ,\protect\sigma ^{2}). */

2013-08-14T08:59:57Z

‎Calculating probabilities when X\sim N(\protect\mu ,\protect\sigma ^{2}).

Rb: /* A few elementary properties: Z\sim N(0,1) */

2013-08-14T08:59:33Z

‎A few elementary properties: Z\sim N(0,1)

Rb: Created page with " = Introduction to the Normal Distribution = It could be argued that the most important probability distribution encountered thus far has been the [[Probability_DiscreteRV#B..."

2013-08-14T08:59:00Z

Created page with " = Introduction to the Normal Distribution = It could be argued that the most important probability distribution encountered thus far has been the [[Probability_DiscreteRV#B..."

New page

= Introduction to the Normal Distribution =

It could be argued that the most important probability distribution encountered thus far has been the [[Probability_DiscreteRV#Binomial| ''Binomial'' distribution ]] for a discrete random variable monitoring the total number of successes in <math>n</math> independent and identical Bernoulli experiments. Indeed, this distribution was proposed as such by Jacob Bernoulli (1654-1705) in about 1700. However as <math>n</math> becomes large, the Binomial distribution becomes difficult to work with and several mathematicians sought approximations to it using various limiting arguments. Following this line of enquiry two other important probability distributions emerged; one was the ''Poisson'' distribution, due to the French mathematician Poisson (1781-1840), and published in 1837. The other, is the ''normal'' distribution due to De Moivre (French, 1667-1754), but more commonly associated with the later German mathematician, Gauss (1777-1855), and French mathematician, Laplace (1749-1827). Physicists and engineers often refer to it as the ''Gaussian'' distribution. There a several pieces of evidence which suggest that the British mathematician/statistician, Karl Pearson (1857-1936) coined the phrase ''normal distribution''.

Further statistical and mathematical investigation, since that time, has revealed that the normal distribution plays a unique role in the theory of statistics; it is without doubt the most important distribution. We introduce it here, and study its characteristics, but you will encounter it many more times in this, and other, statistical or econometric courses. Briefly the motivation for wishing to study the normal distribution can be summarised in three main points:

* it can provide a good approximation to the binomial distribution
* it provides a natural representation for many ''continuous'' random variables that arise in the social (and other) sciences
* many functions of interest in statistics give random variables which have distributions closely approximated by the normal distribution.

We shall see shortly that the normal distribution is defined by a particular ''probability density function''; it is therefore appropriate (in the strict sense) for modelling ''continuous'' random variables. Not withstanding this, it is often the case that it provide an adequate approximation to another distribution, even if the original distribution is ''discrete'' in nature, as we shall now see in the case of a binomial random variable.

== The Normal distribution as an approximation to the Binomial distribution ==

Consider a Binomial distribution which assigns probabilities to the total number of successes in <math>n</math> identical applications of the same Bernoulli experiment. For the present purpose we shall use the example of flipping a coin a number of times (<math>n</math>). As we assume that our coin is fair this implies that the probability of success is 0.5, <math>\pi=0.5</math>. Let the random variable of interest be the ''proportion'' of times that a ''HEAD'' appears and let us consider how this distribution changes as <math>n</math> increases:

* If <math>n=3</math>, the possible proportions could be <math>0,\,\,1/3,\,\,2/3\,\,\,or \,\,\,1</math>
* If <math>n=5</math>, the possible proportions could be <math>0,\,\,1/5,\,\,2/5,\,\,3/5,\,\,4/5\,\,\,or\,\,1</math>
* If <math>n=10</math>, the possible proportions could be <math>0,\,\,1/10,\,\,2/10,</math> etc ...

The probability distributions, over such proportions (with <math>\pi=0.5</math>), for <math>n=3</math>, <math>5</math>, <math>10</math> and <math>50</math>, are depicted in the following Figure.

[[File:ProbNormal_binomialapp.jpg|frameless|600px]]

Notice that the ‘bars’, indicating where masses of probability are dropped, get closer and closer together until, in the limit, all the space between them is squeezed out and a bell shaped mass appears, by joining up the tops of every bar we get something that looks like a nice ''probability density function''. As it turns out, this is one case in which we find a ''NORMAL DISTRIBUTION''. At this stage this is a little odd and it will not be obvious that the result of this operation above (i.e. increasing <math>n</math> to <math>\infty</math>), should lead to a special operation. But hej, it does!

Having motivated the normal distribution via this limiting argument, let us now investigate the fundamental mathematical properties of this ''bell-shape''.

= The Normal distribution =

The normal distribution is characterised by a particular probability density function <math>f(x)</math>, the precise definition of which we shall divulge later. For the moment the important things to know about this function are:

* it is bell-shaped
* it tails off to zero as <math>x\rightarrow \pm \infty </math>
* area under <math>f(x)</math> gives probability; i.e., <math>\Pr \left( a<X\leq b\right) =\int_{a}^{b}f(x)dx.</math>

The Normal density function has the classic ''bell'' shape which is shown here

[[File:ProbNormal_normal1.jpg|frameless|600px]]

The specific ''location'' and ''scale'' of the bell depend upon two ''parameters'' (real numbers) denoted <math>\mu </math> and <math>\sigma </math> (with <math>\sigma >0</math>), as depicted in the above Figure. <math>\mu </math> is the Greek letter ''mu'' (with English equivalent ''m'') and <math>\sigma </math> is the Greek letter ''sigma'' with (English equivalent ''s''). Changing <math>\mu </math> relocates the density (shifting it to the left or right) but leaving it’s scale and shape unaltered. Increasing <math>\sigma </math> makes the density ‘fatter’ with a lower peak but fatter tails; such changes are illustrated here

[[File:ProbNormal_normal2.jpg|frameless|600px]]

Furthermore:

* <math>f(x)</math> is symmetric about the value <math>x=\mu ;</math> i.e., <math>f(\mu +c)=f(\mu-c)</math>, for any real number <math>c</math>.
* <math>f(x)</math> has points of inflection at <math>x=\mu \pm \sigma ;</math> i.e., <math>d^{2}f(x)/dx^{2}</math> is zero at the values <math>x=\mu +\sigma </math> and <math>x=\mu -\sigma</math>.

The above describes all the salient mathematical characteristics of the normal ''pdf''. For what it’s worth, although you will not be expected to remember this, the density is actually defined by:

<math>f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left( -\frac{(x-\mu )^{2}}{2\sigma ^{2}}\right) ,\quad -\infty <x<\infty ;\quad -\infty <\mu <\infty ,\,\,\,\,\sigma >0,</math>

and we say that a continuous random variable <math>X</math> has a normal distribution if and only if it has ''pdf'' defined by <math>f(x)</math>, above. Here, <math>\pi </math> is the number ''Pi'' <math>=3.14159..</math>. . In shorthand, we write <math>X\sim N\left( \mu ,\sigma ^{2}\right) </math>, meaning ‘<math>X</math> is normally distributed with location <math>\mu </math> and scale <math>\sigma </math>’. However, a perfectly acceptable alternative is to say ‘<math>X</math> is normally distributed with ''mean'' <math>\mu </math> and ''variance'' <math>\sigma ^{2}</math>’, for reasons which shall become clear in the next section.

An important special case of this distribution arises when <math>\mu =0</math> and <math>\sigma =1</math>, yielding the ''standard normal density''.

== The standard normal density ==

If <math>Z\sim N(0,1)</math>, then the'' pdf ''for <math>Z</math> is written

<math>\phi (z)=\frac{1}{\sqrt{2\pi }}exp(-z^{2}/2),\,\,\,-\infty <z<\infty ,</math>

where <math>\phi </math> is the Greek letter ''phi, ''equivalent to the English <math>f</math>. The ''pdf'',<math>\phi \left( z\right) </math>, is given a special symbol because it is used so often and merits distinction. Indeed, the standard normal density is used to calculate probabilities associated with a normal distribution, even when <math>\mu \neq 0</math> and/or <math>\sigma \neq 1</math> (see below).

= The normal distribution as a model for data =

Apart from its existence via various mathematical limiting arguments, the normal distribution offers a way of approximating the distribution of many variables of interest in the social (and other) sciences. For example, in the [[Probability_Conditional_Exercises|conditional probability exercises]], some statistics were provided from The Survey of British Births which recorded the birth-weight of babies born to mothers who smoked and those who didn’t. The following Figure, depicts the histogram of birth-weights for babies born to mothers who had never smoked. Superimposed on top of that is normal density curve with parameters set at <math>\mu =3353.8</math> and <math>\sigma =572.6</math>.

[[File:ProbNormal_birth.jpg|frameless|600px]]

As can be seen, the fitted normal density does a reasonable job at tracing out the shape of the histogram, as constructed from the data. (I will leave it as a matter of conjecture as to whether the birth-weights of babies born to mothers who smoked are ''normal''.) The nature of the approximation here is that areas under the histogram record the relative frequency, or ''proportion'' in the sample, of birth-weights lying in a given interval, whereas the area under the normal density, over the same interval, gives the ''probability''.

Let us now turn the question of calculating such probabilities associated with a normal distribution.

= Calculating probabilities =

Since <math>f(x)</math> is a ''pdf'', to obtain probabilities we need to think ''area'' which means we have to ''integrate''. Unfortunately, there is no easy way to integrate <math>\phi (z)</math>, let alone <math>f(x)</math>. To help us, however, special (statistical) tables (or computer packages such as EXCEL) provide probabilities about the standard normal random variable <math>Z\sim N(0,1)</math> and they can be used to obtain probability statements about <math>X\sim N(\mu ,\sigma ^{2})</math>.

To develop how this works in practice, we require some elementary properties of <math>Z\sim N(0,1)</math>

== A few elementary properties: <math>Z\sim N(0,1)</math> ==

Firstly, we introduce the ''cdf ''for <math>Z</math>, This functions is denoted <math>\Phi \left( z\right) </math>, where <math>\Phi </math> is the upper case Greek <math>F</math>, and is defined as follows:

<math>\Phi (z)=\Pr (Z\leq z)=\int_{-\infty }^{z}\phi (t)dt,</math>

the area under <math>\phi (.)</math> up to the point <math>z</math>, with <math>\phi (z)=d\Phi (z)/dz</math>. Now, due to symmetry of <math>\phi (z)</math> about the value <math>z=0</math>, it follows that:

<math>\Phi (0)=\Pr (X\leq 0)=1/2</math>

and, in general,

<math>\begin{aligned}
\Phi (-z) &=&\Pr (Z\leq -z) \\
&=&\Pr (Z>z) \\
&=&1-\Pr (Z\leq z) \\
&=&1-\Phi (z).\end{aligned}</math>

The role of symmetry and calculation of probabilities as areas under <math>\phi (z)</math> is illustrated in the following Figure. In this diagram, the area under <math>\phi (z)</math> is divided up into <math>2</math> parts: the area to the left of <math>a</math> which is <math>\Phi (a)</math>; and the area to the right of <math>a</math> which is <math>1-\Phi (a)</math>. These areas add up to <math>1</math>.

[[File:ProbNormal_sym.jpg|frameless|600px]]

== Calculating probabilities when <math>Z\sim N(0,1)</math> ==

Armed with these properties we can now use the “standard normal” table of probabilities. You can download the table with values for <math>\phi(z)</math> from [[media:NormalTable.pdf|here]]. These are exactly the tables which you will be expected to use in any examinations.

The probabilities provided by this table are of the form <math>\Pr (Z\leq z)=\Phi (z)</math>, for values of <math>0<z<\infty </math>. (<math>\Pr (Z\leq z)</math> for values of <math>z<0</math> can be deduced using symmetry.) For example, you should satisfy yourself that you understand the use of the table by verifying that,

<math>\begin{array}{ccc}
\Pr (Z\leq 0]=0.5; & & \Pr (Z\leq 0.5)=0.6915; \\
\Pr (Z\leq 1.96)=0.975; & & \Pr (Z\geq 1)=\Pr (Z\leq -1)=0.1587,\text{ }etc.\end{array}
</math>

The calculation of the probability

<math>\begin{aligned}
\Pr (-1.62 &<&Z\leq 2.2)=\Pr (Z\leq 2.2)-\Pr (Z\leq -1.62) \\
&=&0.986-0.053 \\
&=&0.933\end{aligned}</math>

is illustrated in the following Figure.

[[File:ProbNormal_calcs.jpg|frameless|600px]]

In this diagram, the areas are divided into <math>3</math> mutually exclusive parts: the area to the left of <math>z=-1.62</math>, which equals <math>0.053;</math> the area to the right of <math>z=2.2</math>, which is equal to <math>0.014;</math> and the area in between, which is equal to <math>0.933</math> the required probability.

== Calculating probabilities when <math>X\sim N(\protect\mu ,\protect\sigma ^{2})</math>. ==

We can calculate probabilities associated with the random variable <math>X\sim N(\mu ,\sigma ^{2})</math>, by employing the following results which shall be stated without proof:

* If <math>Z\sim N(0,1)</math>, then <math>X=\sigma Z+\mu \,\,\,\sim \,\,\,N(\mu ,\sigma ^{2})</math>.
* If <math>X\sim N\left( \mu ,\sigma ^{2}\right) </math>, then <math>Z=\frac{X-\mu }{\sigma }\sim N\left( 0,1\right) </math>.

For example, if <math>Z\sim N\left( 0,1\right) </math>, then <math>X=3Z+6\sim N\left( 6,9\right) ;</math> and, if <math>X\sim N\left( 4,25\right) </math>, then <math>Z=\frac{X-4}{5}\sim N\left( 0,1\right)</math>. These results allow us to translate a ''non-standard'' normal distribution to a ''standard'' normal distribution and hence will enable us to use the standard normal probability table to solve probability problems for all normal distributions. If <math>X\sim N(\mu ,\sigma ^{2})</math>, probabilities about <math>X</math> can be obtained from probabilities about <math>Z</math> via the relationship <math>X=\sigma Z+\mu </math>, since we can then write <math>Z=\frac{X-\mu }{\sigma }</math>.

Let <math>X\sim N(\mu ,\sigma ^{2})</math>, with <math>Z=\frac{X-\mu }{\sigma }\sim N\left(0,1\right)</math>, then

<table border="1">

<tr class="odd">
<td align="left"><math>\Pr (a<X\leq b)</math></td>
<td align="left"><math>=</math></td>
<td align="left"><math>\Pr \left( a-\mu <X-\mu <b-\mu \right) ,\quad </math> subtract <math>\mu </math> throughout,</td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><math>=</math></td>
<td align="left"><math>\Pr \left( \frac{a-\mu }{\sigma }<\frac{X-\mu }{\sigma }\leq \frac{b-\mu }{\sigma }\right) ,\quad </math>divide through by <math>\sigma >0</math> throughout</td>
</tr>
<tr class="odd">
<td align="left"></td>
<td align="left"><math>=</math></td>
<td align="left"><math>\Pr \left( \frac{a-\mu }{\sigma }<Z\leq \frac{b-\mu }{\sigma }\right) ,\quad </math>where <math>Z\sim N\left( 0,1\right)</math></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><math>=</math></td>
<td align="left"><math>\Pr \left( Z\leq \frac{b-\mu }{\sigma }\right) -\Pr \left( Z\leq\frac{a-\mu }{\sigma }\right) , </math></td>
</tr>
<tr class="odd">
<td align="left"></td>
<td align="left"><math>=</math></td>
<td align="left"><math>\Phi \left( \frac{b-\mu }{\sigma }\right) -\Phi \left( \frac{a-\mu}{\sigma }\right) </math>.</td>
</tr>

</table>

We thus find that <math>\Pr (a<X\leq b)=\Phi \left( \frac{b-\mu }{\sigma }\right) -\Phi \left( \frac{a-\mu }{\sigma }\right) </math>, and the probabilities on the right hand side are easily determined from Standard Normal Tables. The key of all these probability calculations is to translate the problem into such a form that you only need probabilities of the type <math>\Phi \left( z \right)</math> which you can get from the standard normal table. The following example illustrates the procedure in practice:

<ul>
<li>''Example'': Let <math>X\sim N(10,16)</math>, what is <math>\Pr (0<X\leq 14)</math> ? Here, <math>\mu =10,\sigma =4,a=0,b=14</math>; so, <math>\frac{a-\mu }{\sigma }=-2.5</math> and <math>\frac{b-\mu }{\sigma }=1</math>. Therefore, the required probability is:
<math>\begin{aligned}
Pr(-2.5 &<&Z\leq 1)=\Pr \left( Z\leq 1\right) -\Pr \left( Z\leq -2.5\right)\\
&=&0.8413-0.0062=0.8351.
\end{aligned}</math></li>
<li>''Example'': A fuel is to contain <math>X\%</math> of a particular compound. Specifications call for <math>X</math> to be between <math>30</math> and <math>35</math>. The manufacturer makes a profit of <math>Y</math> pence per gallon where
<math>Y=\left\{
\begin{array}{l}
10,\quad \text{if}\quad 30\leq x\leq 35 \\
5,\quad \text{if}\quad 25\leq x<30\quad or\quad 35<x\leq 40 \\
-10,\quad \text{otherwise}. \end{array}
\right.</math>
If <math>X\sim N(33,9)</math>, evaluate <math>\Pr \left( Y=10,\right) </math> <math>\Pr \left(Y=-10\right) </math> and, hence, <math>\Pr \left( Y=5\right) </math>.
Here, <math>X\sim N\left( 33,9\right) ;</math> i.e., <math>\mu =33</math> and <math>\sigma =3</math>. Now, since <math>\frac{X-33}{3}\sim N(0,1):</math>
<math>\begin{aligned}
\Pr \left( Y=10\right) &=&\Pr \left( 30\leq X\leq 35\right) \\
&=&\Pr \left( \frac{30-33}{3}\leq \frac{X-33}{3}\leq \frac{35-33}{3}\right)\\
&=&\Pr \left( Z\leq 2/3\right) -\Pr \left( Z\leq -1\right) ,\quad \quad \text{where }Z\sim N\left( 0,1\right) \\
&=&\Phi \left( 2/3\right) -\Phi \left( -1\right) \\
&=&\Phi \left( 0.67\right) -\Phi \left( -1\right) \\
&=&0.7486-0.1587 \\
&=&0.5899.
\end{aligned}</math>
Similar calculations show that
<math>\begin{aligned}
\Pr \left( Y=-10\right) &=&\Pr \left( \left\{ X<25\right\} \cup \left\{X>40\right\} \right) \\
&=&1-\Pr \left( 25\leq X\leq 40\right) \\
&=&1-\Pr \left( \frac{25-33}{3}\leq \frac{X-33}{3}\leq \frac{40-33}{3}\right)\\
&=&1-\left\{ \Phi \left( 7/3\right) -\Phi \left( -8/3\right) \right\} \\
&=&1-\left\{ \Phi \left( 2.33\right) -\Phi \left( -2.67\right) \right\} \\
&=&1-0.9901+0.0038 \\
&=&0.0137.
\end{aligned}</math>
Thus, <math>\Pr \left( Y=5\right) =1-0.5899-0.0137=0.3964</math>.</li></ul>

= Footnotes =

@@ Line 209: / Line 209: @@
 = Additional Resources =
-* Fairly formal information on the Normal distribution can be found on [http://mathworld.wolfram.com/NormalDistribution.html | Wolfram MathWorld] and on [http://en.wikipedia.org/wiki/Normal_distribution | Wikipedia]
+* Fairly formal information on the Normal distribution can be found on [http://mathworld.wolfram.com/NormalDistribution.html Wolfram MathWorld] and on [http://en.wikipedia.org/wiki/Normal_distribution Wikipedia]
-* Khan Academy: There is a range of clips on Khan Academy which are useful. This is the [https://www.khanacademy.org/math/probability/statistics-inferential/normal_distribution/v/introduction-to-the-normal-distribution | first] one.
+* Khan Academy: There is a range of clips on Khan Academy which are useful. This is the [https://www.khanacademy.org/math/probability/statistics-inferential/normal_distribution/v/introduction-to-the-normal-distribution first] one.
-* A clip which goes through a few examples of calculating normal probabilities can be found [http://www.youtube.com/watch?v=Ps15UaIKQpU&feature=share&list=PLW7MJJThJQQs3djo1EL6KCRFeCa6wpfYY | here].
+* A clip which goes through a few examples of calculating normal probabilities can be found [http://www.youtube.com/watch?v=Ps15UaIKQpU&feature=share&list=PLW7MJJThJQQs3djo1EL6KCRFeCa6wpfYY here].
-* How to use EXCEL to calculate probabilities is demonstrated [http://www.youtube.com/watch?v=j27Dl-vV9do | here].
+* How to use EXCEL to calculate probabilities is demonstrated [http://www.youtube.com/watch?v=j27Dl-vV9do here].
 = Footnotes =

@@ Line 209: / Line 209: @@
 = Additional Resources =
-* Fairly formal information on the Normal distribution can be found on [http://mathworld.wolfram.com/NormalDistribution.html| Wolfram MathWorld] and on [http://en.wikipedia.org/wiki/Normal_distribution| Wikipedia]
+* Fairly formal information on the Normal distribution can be found on [http://mathworld.wolfram.com/NormalDistribution.html | Wolfram MathWorld] and on [http://en.wikipedia.org/wiki/Normal_distribution | Wikipedia]
-* Khan Academy: There is a range of clips on Khan Academy which are useful. This is the [https://www.khanacademy.org/math/probability/statistics-inferential/normal_distribution/v/introduction-to-the-normal-distribution| first] one.
+* Khan Academy: There is a range of clips on Khan Academy which are useful. This is the [https://www.khanacademy.org/math/probability/statistics-inferential/normal_distribution/v/introduction-to-the-normal-distribution | first] one.
-* A clip which goes through a few examples of calculating normal probabilities can be found [http://www.youtube.com/watch?v=Ps15UaIKQpU&feature=share&list=PLW7MJJThJQQs3djo1EL6KCRFeCa6wpfYY| here].
+* A clip which goes through a few examples of calculating normal probabilities can be found [http://www.youtube.com/watch?v=Ps15UaIKQpU&feature=share&list=PLW7MJJThJQQs3djo1EL6KCRFeCa6wpfYY | here].
-* How to use EXCEL to calculate probabilities is demonstrated [http://www.youtube.com/watch?v=j27Dl-vV9do| here].
+* How to use EXCEL to calculate probabilities is demonstrated [http://www.youtube.com/watch?v=j27Dl-vV9do | here].
 = Footnotes =

@@ Line 212: / Line 212: @@
 * Khan Academy: There is a range of clips on Khan Academy which are useful. This is the [https://www.khanacademy.org/math/probability/statistics-inferential/normal_distribution/v/introduction-to-the-normal-distribution| first] one.
 * A clip which goes through a few examples of calculating normal probabilities can be found [http://www.youtube.com/watch?v=Ps15UaIKQpU&feature=share&list=PLW7MJJThJQQs3djo1EL6KCRFeCa6wpfYY| here].
-* How to use EXCEL to calculate probabilities is demonstrated [http://youtu.be/j27Dl-vV9do| here].
+* How to use EXCEL to calculate probabilities is demonstrated [http://www.youtube.com/watch?v=j27Dl-vV9do| here].
 = Footnotes =

@@ Line 212: / Line 212: @@
 * Khan Academy: There is a range of clips on Khan Academy which are useful. This is the [https://www.khanacademy.org/math/probability/statistics-inferential/normal_distribution/v/introduction-to-the-normal-distribution| first] one.
 * A clip which goes through a few examples of calculating normal probabilities can be found [http://www.youtube.com/watch?v=Ps15UaIKQpU&feature=share&list=PLW7MJJThJQQs3djo1EL6KCRFeCa6wpfYY| here].
 = Footnotes =

@@ Line 207: / Line 207: @@
 <p>Thus, <math>\Pr \left( Y=5\right) =1-0.5899-0.0137=0.3964</math>.</p></li></ul>
-== Additional Resources ==
+= Additional Resources =
 * Fairly formal information on the Normal distribution can be found on [http://mathworld.wolfram.com/NormalDistribution.html| Wolfram MathWorld] and on [http://en.wikipedia.org/wiki/Normal_distribution| Wikipedia]

@@ Line 128: / Line 128: @@
 In this diagram, the areas are divided into <math>3</math> mutually exclusive parts: the area to the left of <math>z=-1.62</math>, which equals <math>0.053;</math> the area to the right of <math>z=2.2</math>, which is equal to <math>0.014;</math> and the area in between, which is equal to <math>0.933</math> the required probability.
-== Calculating probabilities when <math>X\sim N(\protect\mu ,\protect\sigma ^{2})</math>. ==
+== Calculating probabilities when <math>X\sim N(\mu ,\sigma ^{2})</math>. ==
 We can calculate probabilities associated with the random variable <math>X\sim N(\mu ,\sigma ^{2})</math>, by employing the following results which shall be stated without proof:

@@ Line 82: / Line 82: @@
 To develop how this works in practice, we require some elementary properties of <math>Z\sim N(0,1)</math>
-== A few elementary properties: <math>Z\sim N(0,1)</math> ==
+== A few elementary properties <math>Z\sim N(0,1)</math> ==
 Firstly, we introduce the ''cdf ''for <math>Z</math>, This functions is denoted <math>\Phi \left( z\right) </math>, where <math>\Phi </math> is the upper case Greek <math>F</math>, and is defined as follows: