Lesson 1

The mean is seemingly the most natural property of a random variable. When we want to describe a set of numbers with a single number, we most often use the mean value: everyone is familiar with expressions like “average salary,” “average temperature,” and similar. A random variable is not just a set of numbers, but a set of numbers with weights (in our discrete case), so the mean will also be weighted. Formally, it is called the expected value and is defined as follows: \[\operatorname{E}[\alpha]=\sum_{u \in U}\alpha(u)\cdot \Pr[u] \ .\] It is easy to verify that this is the same as \[\operatorname{E}[\alpha]=\sum_{a \in \mathbb{R}}a\cdot \Pr[\alpha=a] \ .\] (Again, summation is carried out only over those \(a \in \mathbb{R}\) that the random variable \(\alpha\) can take, so such a sum is valid.) For example, if \(\alpha\) is the number appearing on a die, then \[\operatorname{E}[\alpha]=\sum_{i=1}^{6}i \cdot \frac{1}{6}= 3.5 \ .\]