Random Variables¶
Introduction to Random Variables¶
A random variable \(X\) is a function or a mapping from the sample space \(\Omega\) to the real numbers, i.e., it associates a real number to every possible outcome. Mathematically, we say that
We denote a random variable by a capital letter, e.g., \(X\), and its numerical values by a small letter, e.g., \(x\). A function of one or several random variables is also a random variable. For example, \(X+Y\) is a random variable whose value is \(x+y\) when \(X=x\) and \(Y=y\). There can be several random variables defined on a single sample space.
A random variable can be discrete or continuous. Consider tossing a coin 5 times. We can define a random variable as the number of heads that come up. In this case, we have a discrete random variable. We might also consider the height of students in a university, in which case we have a continuous random variable.
Discrete Random Variables¶
A discrete random variable takes countably many values. We say that a set is countable if we can put the elements into a one-to-one correspondence with the set of integers.
The probability law of a discrete random variable is its probability mass function; a continuous random variable is instead described by its probability density function.