A random variable is a numerical quantity that takes on different values based on the outcome of a random event. It is often denoted by the letter X and can be discrete or continuous.
The "break a stick" example is a simple experiment where a stick is broken into two pieces at a random point. The lengths of the two pieces can be represented by a random variable, with the possible values being any number between 0 and the length of the original stick.
A probability distribution is used to describe the likelihood of different values occurring for the random variable in the "break a stick" example. This can be visualized with a graph or table, where the x-axis represents the possible values and the y-axis represents the probability of each value occurring.
The expected value in the "break a stick" example is the average value that we would expect to get if we repeated the experiment many times. It is calculated by multiplying each possible value by its corresponding probability and summing all of these products.
The "break a stick" example can be applied in real life situations where there is uncertainty or randomness involved. For example, it can be used to model the lengths of broken bones in a medical study or the lengths of cracks in a material under stress.