# Probability Distributions [09/08/2015]

In this lecture, we’ll talk about several interesting probability distributions. I assume that perhaps you have already heard of those distributions before, but what I’m going to tell you is how those distributions come into being. After you know how those distributions come into being, you will not feel those distributions as mysterious. Another message that I want to tell you is that what I find is that engineering students, including me many, many years ago, normally just pick a distribution and use this distribution for random, without thinking about what is the connotation of such probability distribution. I hope that after you know how those probability distributions that I talk about in this lecture come into being, you will think about how the other distributions come into being. Then it will help you reason about the distributions that you will work with in the future.

### Distributions from repeated Bernoulli trials

Let us tossing a coin repeatedly. Whenever we toss a coin, we conduct a Bernoulli trial, so we have, as a result, we could have a head up or we could have a tail up. We assign probability $p$ to the event that we get a head up. We assign probability $1-p$ to the events that we get a tail up. We call a head up as a success. We call a tail up as a fail. If we toss this coin for many, many times then we get interesting random variables and distributions. Formally speaking, let us use random variable $X$ to denote the result of a Bernoulli trial. $X$ can take two values: $X$ takes 1 with probability $p$ ($P(X=1)=p$) and $X$ takes 0 with probability $1-p$ ($P(X=0)=1-0)$.

What is the mean value and variance of this Bernoulli trial?