Content

 Introduction

 What is probability?

 Combining probabilities

 The two-state system

 Combinatorial analysis

 The binomial distribution

 The mean, variance, and standard deviation

What is meant by the mean or average of a quantity? Well, suppose that we wanted to calculate the average age of undergraduates at UT Austin. We could go to the central administration building and find out how many eighteen year-olds, nineteen year-olds, etc. were currently enrolled. We would then write something like
\begin{displaymath} {\rm Average~Age} \simeq \frac{N_{18}\times 18 + N_{19}\times 19 +N_{20} \times 20+\cdots} {N_{18}+N_{19}+N_{20}\cdots}, \end{displaymath} (24)

where $N_{18}$ is the number of enrolled eighteen year-olds, etc. Suppose that we were to pick a student at random and then ask ``What is the probability of this student being eighteen?'' From what we have already discussed, this probability is defined
\begin{displaymath} P_{18} = \frac{N_{18}}{N_{\rm students}}, \end{displaymath} (25)

where $N_{\rm students}$ is the total number of enrolled students. We can now see that the average age takes the form
\begin{displaymath} {\rm Average~Age} \simeq P_{18}\times 18 + P_{19}\times 19 + P_{20}\times 20 +\cdots. \end{displaymath} (26)

Well, there is nothing special about the age distribution of students at UT Austin. So, for a general variable $u$, which can take on any one of $M$ possible values $u_1$, $u_2, \cdots, u_M$, with corresponding probabilities $P(u_1)$, $P(u_2),\cdots, P(u_M)$, the mean or average value of $u$, which is denoted $\bar{u}$, is defined as

\begin{displaymath} \bar{u} \equiv \sum_{i=1}^{M} P(u_i)\, u_i. \end{displaymath} (27)

Suppose that $f(u)$ is some function of $u$. Then, for each of the $M$ possible values of $u$ there is a corresponding value of $f(u)$ which occurs with the same probability. Thus, $f(u_1)$ corresponds to $u_1$ and occurs with the probability $P(u_1)$, and so on. It follows from our previous definition that the mean value of $f(u)$ is given by

\begin{displaymath} \overline{f(u)} \equiv \sum_{i=1}^{M} P(u_i)\, f(u_i). \end{displaymath} (28)

Suppose that $f(u)$ and $g(u)$ are two general functions of $u$. It follows that

\begin{displaymath} \overline{f(u)+g(u)} = \sum_{i=1}^{M}P(u_i)\,[f(u_i)+g(u_i)] = \sum_{i=1}^{M}P(u_i)\,f(u_i)+ \sum_{i=1}^{M}P(u_i)\,g(u_i), \end{displaymath} (29)

so
\begin{displaymath} \overline{f(u)+g(u)}= \overline{f(u)}+\overline{g(u)}. \end{displaymath} (30)

Finally, if $c$ is a general constant then it is clear that

\begin{displaymath} \overline{c \,f(u)} = c\,\overline{f(u)}. \end{displaymath} (31)

We now know how to define the mean value of the general variable $u$. But, how can we characterize the scatter around the mean value? We could investigate the deviation of $u$ from its mean value $\bar{u}$, which is denoted

\begin{displaymath} {\mit\Delta} u \equiv u- \bar{u}. \end{displaymath} (32)

In fact, this is not a particularly interesting quantity, since its average is obviously zero:
\begin{displaymath} \overline{{\mit\Delta} u} = \overline{(u-\bar{u})} = \bar{u}-\bar{u} = 0. \end{displaymath} (33)

This is another way of saying that the average deviation from the mean vanishes. A more interesting quantity is the square of the deviation. The average value of this quantity,
\begin{displaymath} \overline{({\mit\Delta} u)^2} = \sum_{i=1}^M P(u_i)\,(u_i - \bar{u})^2, \end{displaymath} (34)

is usually called the variance. The variance is clearly a positive number, unless there is no scatter at all in the distribution, so that all possible values of $u$ correspond to the mean value $\bar{u}$, in which case it is zero. The following general relation is often useful
\begin{displaymath} \overline{(u-\bar{u})^2} = \overline{( u^2-2u\,\bar{u}+\bar{u}^2)}= \overline{u^2}-2\bar{u}\bar{u}+\bar{u}^2, \end{displaymath} (35)

giving
\begin{displaymath} \overline{(u-\bar{u})^2}= \overline{u^2}-\bar{u}^2. \end{displaymath} (36)

The variance of $u$ is proportional to the square of the scatter of $u$ around its mean value. A more useful measure of the scatter is given by the square root of the variance,
\begin{displaymath} {\mit\Delta}^\ast u = \left[\overline{({\mit\Delta} u)^2}\right]^{1/2}, \end{displaymath} (37)

which is usually called the standard deviation of $u$. The standard deviation is essentially the width of the range over which $u$ is distributed around its mean value $\bar{u}$.

 Application to the binomial distribution

 The Gaussian distribution

 The central limit theorem