Probability Theory, 2nd Edition

3 Random variables and their distribution

3.1 Transformation of random values

Assume the probability space $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathcal{A}\mathrm{,}\mathbb{P}\mathrm{)}$ describes a certain random experiment, for example, rolling a die or tossing a coin. If the experiment is executed, a random result $\mathit{\omega }\in \mathrm{\Omega }$ shows up. In a second step, we transform this observed result via a mapping $\mathit{X}:\mathrm{\Omega }\to \mathbb{R}$. In this way we obtain a (random) real number $\mathit{X}\mathrm{(}\mathit{\omega }\mathrm{)}$. Let us point out that X is a fixed, nonrandom function from Ω into $\mathbb{R}$; the randomness of $\mathit{X}\mathrm{(}\mathit{\omega }\mathrm{)}$ stems from the input $\mathit{\omega }\in \mathrm{\Omega }$.

Example 3.1.1.

Toss a fair coin, labeled on one side with “0” and on the other side with “1,” exactly n times. The appropriate probability space is $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathcal{P}\mathrm{(}\mathrm{\Omega }\mathrm{)}\mathrm{,}\mathbb{P}\mathrm{)}$, where $\mathrm{\Omega }={\{0\mathrm{,}1\}}^{\mathit{n}}$ and $\mathbb{P}$ is the uniform distribution on Ω. The result of the experiment ...