12 Lecture 11, January 31, 2024

12.1 Distinction of the definition “discrete” of the sample space and the random variable

Recall the definition of the Range of a random variable from last lecture:

Definition 12.1 (Range) The values that a random variables takes is called the range of the random variable. We often denote the range of a random variable \(X\) by \(X(S)\).

  • We say that, a random variable is discrete if its range \(X(\omega)\) is discrete (finite or countable, in another word, we can say it is at most countable).

  • We say, a sample space \(S\) is discrete if \(S\) is finite or countable.

The sample space \(S\) and the range of the random variable are two different things, so do not get confused! We can have a discrete random variable while the sample space \(S\) is continuous!

12.2 Cumulative distribution function

Definition 12.2 (Range) The cumulative distribution function (cdf) of a random variable \(X\) is \[ F_X(x) = P(X \le x),\;\; x \in {\mathbb{R}}. \]

Note: The cumulative distribution function \(F_X\) is always defined over the entire real line \(\mathbb{R}\), while the probability function may not always be defined! Hence, the cumulative distribution function is an useful tool! (but do not worry about it now.)

Notation: \(F_X(x) = P(X\le x) = P(\{\omega \in S : X(\omega)\in x\}).\)

If \(X\) is discrete with probability function \(f_X\) (i.e. if \(f_X\) exists), then we can calculate the cdf from summing up the pdf as \[ F_X(x)=P( X \le x ) = \sum_{y:\; y\le x}f_X(y). \]

12.2.1 Properties of the cumulative distribution function

Let \(F_X(\cdot)\) be a cdf. Then the following holds

  1. \(F_X(x)\in [0,1]\)

  2. \(F_X(x) \le F_x(y)\) whenever \(x<y\) (i.e. \(F_X(\cdot)\) is a non-increasing function.)

  3. \(\lim\limits_{x \to - \infty } F_X(x)=0\), and \(\lim\limits_{x \to \infty } F_X(x) = 1\).

  4. \(F_X\) is right continuous, i.e., \(F(x_0)=\lim_{x\downarrow x_0} F(x)\) for all \(x_0\in\mathbb{R}\).

12.3 Special distributions with names

12.3.1 Discrete uniform distribution

The first named distribution we look at is the discrete uniform distribution

Definition 12.3 (Discrete uniform distribution) uppose the range of the random variable \(X\) is \(\{a,a+1,\dots, b\}\), where \(a,b\in\mathbb{Z}\), and suppose all values are equally likely. Then we say that \(X\) has a discrete uniform distribution on \(\{a,a+1,\dots,b\}\), shorthand: \(X \sim U[a,b]\).

12.3.1.1 Probability function and distribution function

If \(X \sim U[a,b]\), then its probability function is given by \[ f_X(x)= P(X = x) = \begin{cases} \frac{1}{b - a + 1}, &\quad\text{ if }x \in\{a,a+1,\dots,b\}, \\ 0, &\quad\text{ otherwise} \end{cases} \] and corresponding (cumulative) distribution function is \[ F_X(x)= P(X \leq x) = \begin{cases} 0, &\quad\text{ if } x<a\\ \frac{\lfloor x\rfloor - a + 1}{b - a + 1}, &\quad\text{ if }x \in\{a,a+1,\dots,b\}, \\ 1, &\quad\text{ if } x\geq b,\end{cases} \] where \(\lfloor x\rfloor=\max\{z\in\mathbb{Z}: z\leq x\}\) is the floor function.