Discrete Probability Distributions

This page contains a list of all the discrete distributions done in class. In each an attempt has been made to identify the history and why thy name. One of them still remains elusive. More links and adjustments to this page are in the pipeline as the semester progresses.

Bernoulli Distribution

Experiment obeys:
(1) a single trial with two possible outcomes (success and failure)
(2) P({ trial is successful })=p
Random variable $X$: number of successful trials (zero or one)
Probability mass function:

\begin{displaymath}pmf(x) =(defined)= P(X=x)= p^x(1-p)^{n-x}\end{displaymath}

Mean and variance: $\mu=p$, $\sigma^2=p(1-p)$
Example: tossing a fair coin once
Why its name : This random variable was invented by one of the Bernoulli brothers. Not sure how the history goes. But here is a look at the Jacob Bernoulli and family.


Binomial Distribution

Experiment obeys:
(1) $n$ repeated trials
(2) each trial has two possible outcomes (success and failure)
(3) P({ trial is successful })=p
(4) the trials are independent
Random variable X: number of successful trials
Probability mass function:

\begin{displaymath}pmf(x)=(defined)=P(X=x)= {\textstyle{n\choose x}}p^x(1-p)^{n-x}\end{displaymath}

Mean and variance: $\mu=np$, $\sigma^2=np(1-p)$

Example: tossing a fair coin n times

Approximations: (1) binomial(x;n,p)~ poisson(x;=pn) if $p«1, x« n$

Why its name : The name derives itself from the coefficents of the Binomial expansion (a + b)^n with a being p and b being 1-p. The expansion was contributed by Blaise Pascal( the page has a link for the definition of binomial theorem, expansion and pascals triangle ). The coefficients are the numbers in Psacal's Triangle.



Geometric Distribution

Experiment obeys:
(1) indeterminate number of repeated trials
(2) each trial has two possible outcomes (success and failure)
(3) $P\Big(\big\{i^{\rm th} \hbox{ trial is successful}\big\}\Big)=p$ for all $i$
(4) the trials are independent
Random variable: trial number of first successful trial
Probability mass function:

\begin{displaymath}pmf(x) = p (1-p)^{x-1} \end{displaymath}

Mean and variance: $\mu=\frac {1}{p}$, $\sigma^2=\frac {1-p}{p^2}$

Example: repeated attempts to start an engine, or playing a lottery until you win

Why its name : The name derives itself from the geometric progression. That is a series is called a geometric progression if consecutive terms differed by a common ratio. In the above case the common ratio between any two probabilities is p. Euclid of Alexandria ( has a link for the definition of geometric progression )is credited with the notion of geometric progression. I am really not sure though.



Negative Binomial Distribution

Experiment obeys:
(1) indeterminate number of repeated trials
(2) each trial has two possible outcomes (success and failure)
(3) $P\Big(\big\{i^{\rm th} \hbox{ trial is successful}\big\}\Big)=p$ for all $i$
(4) the trials are independent
(5) keep going until $r^{\rm th}$ success
Random variable: trial number on which $r^{\rm th}$ success occurs
Probability mass function:

\begin{displaymath}pmf(x) = {x-1\choose r-1}p^r (1-p)^{x-r}\end{displaymath}


Mean and variance: $\mu=\frac {r}{p}$, $\sigma^2=\frac {r(1-p)}{p^2}$

Example: fabricating $r$ nondefective computer chips
Why its name :


Poisson Distribution

Experiment obeys: count the number of occurrences of some event in a specified time interval or in a specified region of space where:
(1) the events occur at a point in time or space
(2) the number of events occurring in one region is independent of the number
occurring in any disjoint region
(3) the probability of more than one event occurring at the same point is negligible
(4) the probability of $n$ events in region #1 is the same as the probability of $n$ events in region #2, when the regions have the same size
Random variable: number of events occurring in the given time interval or
region of space
Probability mass function:

\begin{displaymath}pmf(x) = \frac {e^{-\lambda }\lambda ^x}{x!} \end{displaymath}

where $\lambda $ is the average number
of events in the given region

Mean and variance: $\mu=\lambda $, $\sigma^2=\lambda $

Example: telephone calls arriving at a switchboard in a specified one hour period
Why its name: Recherchés sur la probabilité des jugements is one of the important works of Siméon Denis Poisson. In this he laid the ground work for the distribution now known as Poisson.



Hypergeometric Distribution

Experiment obeys:
(1) a random sample of size $n$ is selected from $N$ items
(2) there are $k$ items of one type (called successes) and $N-k$ items of another type (called failures)
Random variable: number of successes selected
Probability mass function:

\begin{displaymath}pmf(x)= \frac{{k\choose x}{{N-k\choose n-x}}}{{N\choose n}} \end{displaymath}


Mean and variance: $\mu=n\frac {k}{N}$, $\sigma^2=\frac {N-n}{N-1}n\frac {k}{N}\big(1-\frac {k}{N}\big)$

Example: selecting a random sample of 5 spark plugs from a batch of 40 of which 3 are defective

Why its name: Hyper-geometric motion is a type of motion that does not posses regular geometric properties. (by regular we mean : if a sphere were set spinning it would rotate with a regular periodicity, and not in some jerky, non-geometric way.) But there are many instances in the universe where this is not true. Johann Carl Friedrich Gauss introduced functions( the page has a link for the definition of hypergeometric function ). that described such motions in his work "Disquisitiones generales circa seriem infinitam". The probability mass function for the above experiment resembles this function.


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