Lesson 1

As we have already discussed, there are two natural ways to define a probability distribution: by separately describing the probability of each outcome or by describing a random process/experiment. When working with random variables, another method appears: defining the probability density function or the cumulative distribution function of a random variable.

The probability density function of a random variable \(\alpha \colon U \to \mathbb{R}\) is a function \(\operatorname{PDF}_{\alpha}\colon \mathbb{R}\to [0,1]\), defined as follows: \[\operatorname{PDF}_{\alpha}(x)=\Pr[\alpha=x] \ .\] Of course, \[\sum_{x}\operatorname{PDF}_{\alpha}(x)=1 \ \] (the sum here is taken over the set of values taken by the random variable; this set is at most countable, so everything is valid). The cumulative distribution function of a random variable \(\alpha\) is a function \(\operatorname{CDF}_{\alpha}\colon \mathbb{R} \to [0,1]\), defined as: \[\operatorname{CDF}_{\alpha}(x)=\Pr[\alpha \le x]=\sum_{z \le x}\operatorname{PDF}_{\alpha}(z) \ .\]