Discrete Random Variables

Let $S$$\large S$ be the sample space associated with some experiment $E$$\large E$. A random variable $X$$\large X$ is a function that assigns a real number $X(s)$$\large X(s)$ to each elementary event $s\in S$$\large s \in S$. In this case, as the outcome of the experiment, i.e. the specific $s$$\large s$, is not predetermined, then the value of $X(s)$$\large X(s)$ is not fixed. This means that the value of the random variable is determined by the specific outcome of the experiment.

A discrete random variable is one that can take on at most a countable number of possible values.

Probability Mass Function

For a discrete random variable $X$$\large X$, the probability mass function $p(a)$$\large p(a)$ of $X$$\large X$ is defined as:

Properties

• $\sum _{i}p(x_i)=\sum _{i}p\{X=x_i\}=1$$\large \sum\limits_i p(x_i) = \sum\limits_i p\{X = x_i\} = 1$

Cumulative Distribution Function

For a discrete random variable $X$$\large X$, the cumulative distribution function $F(x)$$\large F(x)$ is defined as:

Properties

• $F(a)=\sum _{allx\le a}p(x)$$\large F(a) = \sum\limits_{all \: x \le a} p(x)$
• $a<b⟹F(a)\le F(b)$$\large a \lt b \implies \Large F(a) \le F(b)$
• $\underset{a\to \mathrm{\infty }}{lim}F(a)=1$$\large \lim\limits_{a \to \infty} F(a) = 1$
• $\underset{a\to -\mathrm{\infty }}{lim}F(a)=0$$\large \lim\limits_{a \to -\infty} F(a) = 0$
• $\underset{n\to \mathrm{\infty }}{lim}F(a_n)=F(a)$$\large \lim\limits_{n \to \infty} F(a_n) = F(a)$ for any decreasing sequence $a_n$$\Large a_n$ that converges to $a$$\large a$
• $P\{a<X\le b\}=F(b)-F(a)$$\large P\{a \lt X \le b\} = F(b) - F(a)$

Expected Value

For a discrete random variable $X$$\large X$, the expected value $E[X]$$\large E[X]$ is defined as:

Expected Value of a Function of a Random Variable

For a real-valued function $Y=g(X)$$\large Y = g(X)$ of a random variable $X$$\large X$, the expected value $E[Y]$$\large E[Y]$ is defined as:

Thus we can show:

where $a$$\large a$ and $b$$\large b$ are constants.

Variance

For a discrete random variable $X$$\large X$, the variance $Var(X)$$\large Var(X)$ is defined as:

where $\mu =E[X]$$\large \mu = E[X]$.

Thus we can show:

where $a$$\large a$ and $b$$\large b$ are constants.

Standard Deviation

For a discrete random variable $X$$\large X$, the standard deviation $SD(X)$$\large SD(X)$ is defined as:

Bernoulli Random Variable

The Bernoulli random variable $X$$\large X$, corresponds to an event $E$$\large E$ which have two possible outcomes, $X=0$$\large X = 0$ and $X=1$$\large X = 1$, and its probability mass function is defined as:

where $0\le p\le 1$$\large 0 \le p \le 1$.

Properties

• $E[X]=p$$\large E[X] = p$
• $Var(X)=p(1-p)$$\large Var(X) = p(1 - p)$
• $SD(X)=\sqrt{p(1-p)}$$\large SD(X) = \sqrt{p(1 - p)}$

Binomial Random Variable

The Binomial random variable $X$$\large X$, denotes the total number of times that event $E$$\large E$ occurs when we repeat the Bernoulli experiment $n$$\large n$ independent times, and its probability mass function is defined as:

where $i=0,1,...,n$$\large i = 0,\:1,\:...,\: n$.

Properties

• $E[X]=np$$\large E[X] = np$
• $Var(X)=np(1-p)$$\large Var(X) = np(1 - p)$
• $SD(X)=\sqrt{np(1-p)}$$\large SD(X) = \sqrt{np(1 - p)}$

Poisson Random Variable

The Poisson random variable $X$$\large X$, denotes the total number of times that event $E$$\large E$ occurs when we repeat the Bernoulli experiment continuously during a period of time, and its probability mass function is defined as:

where $\lambda$$\large \lambda$ is the expected value of the outcome in the given period of time.

Properties

• $E[X]=\lambda$$\large E[X] = \lambda$
• $Var(X)=\lambda$$\large Var(X) = \lambda$
• $SD(X)=\sqrt{\lambda }$$\large SD(X) = \sqrt{\lambda}$

Geometric Random Variable

Geometric distribution is the probability distribution of the geometric random variable $X$$\large X$ that denotes the total number of times we have to repeat the Bernoulli experiment until event $E$$\large E$ occurs, and its probability mass function is defined as:

Properties

• $E[X]=\frac{1}{p}$$\large E[X] = \frac{1}{p}$
• $Var(X)=\frac{1-p}{p^2}$$\large Var(X) = \frac{1 - p}{p^2}$
• $SD(X)=\frac{\sqrt{1-p}}{p}$$\large SD(X) = \frac{\sqrt{1 - p}}{p}$

Negative Binomial Random Variable

Negative binomial distribution is the probability distribution of the negative binomial random variable $X$$\large X$ that denotes the total number of times we have to repeat the Bernoulli experiment until event $E$$\large E$ occurs $r$$\large r$ times, and its probability mass function is defined as:

Properties

• $E[X]=\frac{r}{p}$$\large E[X] = \frac{r}{p}$
• $Var(X)=\frac{r(1-p)}{p^2}$$\large Var(X) = \frac{r(1 - p)}{p^2}$
• $SD(X)=\frac{\sqrt{r(1-p)}}{p}$$\large SD(X) = \frac{\sqrt{r(1 - p)}}{p}$

Hypergeometric Random Variable

Consider $N$$\large N$ items numbered $1,2...,N$$\large 1,\:2\:...,\:N$ and that $n$$\large n$ items where chosen at random from the $N$$\large N$ items. Then, hypergeometric distribution is the probability distribution of the hypergeometric random variable $X$$\large X$ that denotes the total number of times we choose chose items that are numbered $1,2...,m$$\large 1,\:2\:...,\:m$, where $m<n$$\large m < n$, and its probability mass function is defined as:

Properties

• $E[X]=\frac{nm}{N}$$\large E[X] = \frac{nm}{N}$
• $Var(X)=\frac{N-n}{N-1}np(1-p)$$\large Var(X) = \frac{N - n}{N - 1} np(1 - p)$
• $SD(X)=\sqrt{\frac{N-n}{N-1}np(1-p)}$$\large SD(X) = \sqrt{\frac{N - n}{N - 1} np(1 - p)}$