Application of Statistics in Physics

Claimed by Edwin Solis (April 16th, Spring 2022)

With the development of Quantum Mechanics and Statistical Mechanics, the subject of Statistics has become quintessential for understanding the foundation of these physical theories.

Basics

Probability

Probability is the numerical description of the likelihood of an event occurring from a sample space written as a value between 0 and 1. This event is just the outcome of executing an experiment, and the sample space is just the whole set of outcomes possible from this experiment. Usually, the probability for equally likely outcomes is written mathematically as

[math]\displaystyle{ P(A)=\frac{\text{# Number of times A occurs}}{\text{# Total number of outcomes}} }[/math]

For example, for a six face fair dice, we have a sample space [math]\displaystyle{ S=\{1, 2, 3, 4, 5, 6\} }[/math], where the probability of obtaining a 3 from a dice throw is given by [math]\displaystyle{ P(X=3)=\frac{1}{6} }[/math]. Here the dice throw is the experiment while the event is the outcome of the number obtained from the dice.

More generally, however, we can define a probability space for an experiment with discrete outcomes as a mathematical function called probability mass function (pmf). In the dice example, the pmf would be written as

[math]\displaystyle{ \begin{align} P(X=x) = \begin{cases} \frac{1}{6} &\;| 1 \leq x \leq 6\\ 0 &\;| otherwise \end{cases} \end{align} }[/math]