Independent repetitions of an experiment
One important situation that leads to independent random variables is when some random experiment is repeated in essentially the same way.
If the experiment is repeated twice, it is usually reasonable to assume that the resulting two variables are not only independent, but also both have the same distribution. This allows us to dispense with the subscripts for their probability functions,
\[ p_X(\cdot) \;=\; p_Y(\cdot) \;=\; p(\cdot) \]If this is extended to \(n\) independent repetitions of a random experiment, we get \(n\) independent identically distributed random variables. These are often abbreviated to iidrv's and are also called a random sample from the distribution with probability function \(p(x)\).
Definition
A random sample of \(n\) values from a distribution is a collection of \(n\) independent random variables, each of which has this distribution.
Random samples often arise in statistics, and the following theorem is central to their analysis.
Probabilities for random samples
The probability that the values in a discrete random sample are \(x_1, x_2, ..., x_n\) is
\[ P(X_1 = x_1, X_2 = x_2, ..., X_n = x_n) \;\; = \;\; \prod_{i=1}^n p(x_i) \](Proved in full version)