Appendix A - Basic Probability Concepts

In this appendix definitions and some example calculations are presented which will aid in our discussions. This is not meant to be a comprehensive introduction to the topic. It is primarily meant to serve as a means for introducing notation and terminology for the course. This example is a variation of one given by David Griffiths in Intoduction to Quantum Mechanics (David J. Giffiths’s book).

Example: Suppose that in some room, there are four people. Their heights in meters are:

1. 1 person is 1.5 meters tall
2. 1 person is 1.6 meters tall
3. 2 people are 1.8 meters tall

LetFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\,\!} will stand for the number of people. We might write their heights as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(1.5) = 1\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(1.6)=1\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(1.8)=2\,\!} and the total number of people is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \sum_{j=0}^\infty N(j), \,\!}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\,\!} runs over all values and in this case we are rounding to the nearest tenth of a meter. Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\,\!} =4 of course.

Now if I draw a name out of a hat which contains each person's name once, I will get the person's name which is 1.6 meters tall with probability $1/4\,\!$. (We assume that each person has a unique name and that it appears once and only once in the hat.) We write this as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(1.6) = 1/4, \,\!}

and we would generally write for any value

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(j) = \frac{N(j)}{N}. \,\!}

Now since we are going to get someone's name when we draw, we must have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_j P(j) = 1, \,\!}

which is easy enough to check.

There are several aspects of this probability distribution that we might like to know. Here are some which are particularly useful:

1. The most probable values for the height is 1.8 meters.
2. The median is 1.7 meters (two people below, and two above)
3. The average (or mean) is given by

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\left\langle height\right\rangle &= \frac{1(1.5)+1(1.6)+2(1.8)}{4} \\ &= \frac{6.8}{4} = 1.7 \end{align}}	(A.1)

Note that the mean and the median do not have to be the same. The median is the middle number in the list, if there is an odd number, otherwise it is the mean of the two in the middle. These two just happen to be the same here. The bracket Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\cdot\right\rangle\,\!} is fairly standard notation and it means generally, that we obtain the average value of a function by calculating

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle f(j)\right\rangle = \sum_{j=0}^\infty f(j)P(j).\,\!}

For the average this is just

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle j\right\rangle = \sum_{j=0}^\infty jP(j)= \sum_{j=0}^\infty j\frac{N(j)}{N}. \,\!}

Note: The average value is called the expectation value in quantum mechanics. This can be misleading becase it is not the most probable and it is not ''what to expect.''

When one would like to discuss a properties of a particular probability distribution, describing it takes some effort. It is not enough to know the average, median, and most probable values. This leaves a lot of details of the probability distribution unknown to us if these are all we are given. What else would one like to know? Without describing it entirely, one may like to know more about the ''shape'' of the distribution. For example, how spread out is it?

The most important measure of this is the variance,, which is the standard deviation squared ($\sigma ^{2},\!$). The variance is defined as (in terms of our variable $j\,\!$)

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2 = \langle(\Delta j)^2\rangle, \,\!}	(A.2)

where $\Delta j=j-\langle j\rangle \,\!$. This can also be written as

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2 = \langle j^2\rangle - \langle j \rangle^2.\,\!}	(A.3)