Random Variables

Discrete Uniform distribution

Each value has the same probability of happening. X is uniformly distributed if and only if

$$\mathbb{P}(X=x_i)=\frac{1}{n}\mathrm{\forall }i=1,\dots ,n$$

Bernoulli

It can only take two possible values, usually 0 and 1. This random variable models random experiments that have two possible outcomes, sometimes referred to as “success” and “failure”.

A random variable $X$ is a Bernoulli random variable with parameter $p$, where $p$ is the probability of success, shown as $X\sim Bernoulli(p)$, if

$$\mathbb{P}(X=x_i)=\{\begin{array}{ll}p,& \text{ for }x=1\\ 1-p,& \text{ for x = 0}\\ 0,& \text{ otherwise}\end{array}$$

Example: A die is tossed, and we are interested at die showing 5, so it either it shows a 5 or not.

Binomial distribution

Assume that we repeat a Bernoulli experiment n times, also, assume that the repetitions are independent, i.e., the result of an experiment won’t impact another one. A random variable $X$ is said to be a binomial random variable with parameters $n$ and $p$, shown as $X\sim Binomial(n,p)$, if

$$\mathbb{P}(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k},k=0,1,...,n$$

Example: A coin is thrown 3 times. What is the probability of 2 heads?

Geometric distribution

Repeating independent Bernoulli trials until observing the first success. A random variable $X$ is said to be a geometric random variable with parameter $p$, shown as $X\sim Geometric(p)$ if

$$\mathbb{P}(X=k)=p(1-p{)}^{k-1},k=1,2,...$$

Poisson distribution

Used in scenarios where we are counting the occurrences of certain events in an interval of time or space. A random variable $X$ is said to be a Poisson random variable with parameter $\lambda$, shown as $X\sim Poisson(\lambda )$, if

$$\mathbb{P}(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!},k=0,1,...$$

Example: A phone number receives, on average, 5 calls per minute. What is the probability of not receiving a call in one minute?

Uniform distribution

A random variable $X$ is said to have a Uniform distribution over the interval $[a,b]$, shown as $X\sim \mathcal{U}(a,b)$, if:

$$f(x;a,b)=\{\begin{array}{ll}\frac{1}{b-a},& \text{if }a\le x\le b\\ 0,& \text{ otherwise}\end{array}$$

Exponential distribution

A random variable $X$ is said to have an exponential distribution with parameter $\lambda >0$, shown as $X\sim Exp(\lambda )$, if

$$f(x;\lambda )=\{\begin{array}{ll}\lambda e^{-\lambda x},& \text{ for }x>0\\ 0,& \text{ otherwise}\end{array}$$

Normal (Gaussian) distribution

The normal distribution is by far the most important probability distribution. One of the main reasons for that is the Central Limit Theorem (CLT) that we will discuss in another moment.

A random variable $X$ is said to have a normal distribution with parameter $\mu \in \mathbb{R}$ and ${\sigma }^2>0$ if

$$f(x;\mu ,{\sigma }^2)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}},-\mathrm{\infty }<x<\mathrm{\infty }$$

Below we can see the representation of a normal distribution with mean $\mu$ and standard deviation $\sigma$

When $\mu =0$ and $\sigma =1$ we say that we have a standard normal distribution ans we represent as $Z\sim \mathcal{N}(0,1)$ and the probability function is reduced to

$$f(z)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{z^2}{2}},-\mathrm{\infty }<z<\mathrm{\infty }$$

If $X\sim \mathcal{N}(\mu ,{\sigma }^2)$, then the variable defined as $Z=\frac{X-\mu }{\sigma }$ has mean 0 and variance 1