Random variable $X$

Source: Wiki

• Alternative name: stochastic variable
• Function that assigns each elementary outcome $\omega$ in the sample set $\Omega$ to another set $\mathcal{A}$ (often to $\mathbb{R}$) $$X: \Omega \rightarrow \mathcal{A}$$
• $P(X = x)$ is the probability that the function $X$ attains the value $x \in \mathcal{A}$
• $x$: specific realisation of a random variable $X$
• Can be discrete (e.g. number of students) or continuous (e.g. height of students)

Example

Source

• Experiment: two dice are thrown
• Sample set: $\left{ (D_i, D_j) \right}$ $\forall, i,j$
• $X$: random variable that gives the number of dice with sixes
• Example trials:
  • Trial 1: (2, 3) –> $X = 0$
  • Trial 2: (6, 1) –> $X = 1$
  • Trial 3: (6, 6) –> $X = 2$