While a random variable is simply a function $$ X:S\to R $$ defined on a probability space, it is the next central object of probability introduced in this course. For the target set there is a lot flexibility but, in this course, we will focus on the case $R=\mathbb{R}$. As far as the intuition and the understanding go, it is important to realize that a random variable can only be understood through the probabilty it has to taken on certain values, like in $$ P\bigl( X\in [a,b]\bigr)\text{ for }a\leq b\in \mathbb{R}. $$ This means that $X$ cannot be assigned a specific value, but only a probability to assume certain values. Let $X$, e.g., denote the outcome of rolling a fair die. Then it is not possible to say which value $X$ has, but one can say that $$ P \bigl( X\in \{1,3,5\}\bigr)=\frac{1}{2} \text{ or }P(X=4)=\frac{1}{6}, $$ for instance. Take a minute to find many examples of everyday quantities that can be viewed as random variables.
An involved example provides further evidence of the importance of conditional probability and the ability to effectively reduce complexity by conditioning.
After introducing random variables, we look at examples and define the so-called cumulative distribution function of a random variable $X$ given by $$ F_X(x)=P(X\leq x),\: x\in \mathbb{R}. $$
An important class of random variables is that of discrete random variables characterized by the fact that they take on at most countably many values. In other words it holds $$ X(S)=\big\{x_1,x_2,\dots\big\}=\big\{ x_k\, |k\in \mathbb{N}\big\} $$ for values $x_k\in \mathbb{R}$, $k\in \mathbb{N}$. To such random variables one can associate the so-called probability mass function $p_X$ defined by $$ p_X(x)=P\bigl( X=x\bigr),\: x\in X(S), $$ which records the probability of assuming specific values. Finally the concept of expectation $$ E(X)=\sum _{x\in X(S)} x\, p_X(x)=\sum _{k\in \mathbb{N}} x_k\, p_X(x_k)=\sum _{k\in \mathbb{N}} x_k\, P\bigl( X=x_k\bigr) $$ is introduced for discrete random variables $X$ and an intuition is offered which clarifies the motivation for the definition.