Random Variables · Random Variables

Lesson 6

Nikolai Chukhin · Alexander S. Kulikov

A random variable (or even several) can be transformed using an arbitrary numerical function to obtain another random variable. For example, let \(U=\{a, b, c, d, e, f\}\), and let the probabilities of the elementary outcomes be defined as follows:

Note that the elementary outcomes are not numbers but simply six objects labeled with letters. To define a random variable \(\alpha \colon U \to \mathbb{R}\), we specify its value for each elementary outcome. For example:

Once the random variable is defined, we can estimate the probability that its value satisfies some property: \[\begin{align*}\Pr[\alpha=2]&=\Pr[\{a, d\}]=0.5 \ .\\ \Pr[\alpha \ge 3]&=\Pr[\{c, e, f\}]=0.3 \ .\\ \Pr[\alpha \bmod 3 = 0]&= \Pr[c] = 0.1 \ .\\\end{align*}\] Finally, since \(\alpha(u)\) is a number, we can apply a real function to it to obtain a new random variable. Suppose, for example, that \(f \colon \mathbb{R}\to \mathbb{R}\) is defined as \(f(x)=x^{2}-3\). Then \(f(\alpha)\) is also a random variable on the same probability space \((U, \Pr)\) with the following values:

Instead of \(f(\alpha)\), we usually write just \(\alpha^{2}-3\). For example, we can consider random variables \(\frac{1}{\alpha}\) and \(e^{\alpha}\). Similarly, we can define functions of several random variables: for example, \(3\alpha\sin \beta + 5\beta^{2}\gamma^{3}\). Clearly, the new random variable has the same type as the original one: \[f(\alpha) \colon U \to \mathbb{R} \ .\]