Statistics
Lecture 6
Discrete probability distributions
•David Bartl
•Statistics
•INM/BASTA

Outline of the lecture
•Discrete probability distributions
•Discrete uniform distribution
•Bernoulli Trials
•Binomial distribution
•Poisson distribution
•Some other discrete probability distributions

Experiment — Trial — Random variable



Random variable — Dataset
•To conclude, the random variable assigns a numerical value to each outcome of the random
experiment.
•Now, a dataset is a collection of measurements and observations, i.e.
it is a collection of data.  A data unit is an entity of the population under study,
and a data item or a variable is a characteristics of each data unit.
We are considering numerical (quantitative) variables now.
•We assume the hypothesis that the data items in the dataset are realizations
of the random variable, i.e. the random variable (via the trials of the random experiment)
generates the data.

Examples of discrete random variables



Discrete probability distributions
•The purpose of this lecture, however, is to present the most important,
yet elementary, discrete probability distributions.
•
•We shall present:
•  —  the uniform distribution
•  —  Bernoulli’s experiment
•  —  the binomial distribution
•  —  Poisson’s distribution

Discrete
uniform distribution
•
•

Uniform distribution (discrete)



Uniform distribution (discrete)



Uniform distribution (discrete)



Uniform distribution (discrete)



Uniform distribution (discrete)



Uniform distribution (discrete)



An application:  the German Tank Problem



Binomial distribution
•Bernoulli Trials
•Binomial distribution
•

Bernoulli Trials



Examples of Bernoulli Trials



Mathematical model of the Bernoulli Trial



Binomial distribution



Binomial distribution



Binomial distribution



Binomial distribution



Binomial distribution



Binomial distribution



Binomial distribution in Excel



Binomial distribution in Excel



Poisson distribution
•
•

Poisson distribution
•There are some events, such as
•  —  customers coming to a shop during one hour (between 10:00 and 11:00, say)
•  —  telephone calls incoming during one hour (between 10:00 and 11:00, say)
•  —  requests incoming to a server during one minute (between 10:00 and 10:01)
•  —  meteorites of diameter  ≥ 1 meter hitting the Earth during a year
•  —  decay events from a radioactive source
•that (as we suppose) have some properties in common.

Poisson distribution
•Suppose that a random event occurs repeatedly and satisfies
the following assumptions:
•the event can occur at any time
•the average number of occurrences of the event during an interval of time
of a fixed length is constant; the number does not depend on the beginning
of the interval, and does not depend on the number of occurrences of the event before the beginning
of the time interval
•the average number of occurrences of the event during an interval of time is proportional to the
length of the interval
•…

Poisson distribution



Poisson distribution



Poisson distribution



Poisson’s Theorem



Poisson distribution



Poisson distribution



Poisson distribution



Poisson distribution



Poisson distribution



Poisson distribution



Poisson distribution in Excel



Poisson distribution:  Examples
•The number of telephone calls received by a call centre per hour.
•The number of customers coming to the shop per hour.
•The number of radioactive decay events per second from a radioactive source.
•The number of clicks per second of a Geiger-Müller counter.
•The number of defaults per year in risk modelling.
•The number of some failures / accidents / … per year.

Some other discrete probability distributions
•Negative binomial distribution
•Lady tasting the tea
•Hypergeometric distribution
•

Negative binomial distribution



Negative binomial distribution



Hypergeometric distribution
•A Lady Tasting the Tea:  We prepare a cup of tea with milk.
•There are two ways to prepare the cup:
•pour the tea into the cup first and then add the milk,
•pour the milk into the cup first and then add the tea.
•
•A lady says that she can recognize by the taste of the tea
how the cup was prepared.

Hypergeometric distribution
•A Lady Tasting the Tea:  We prepare  8  cups of tea with milk.
•We prepare:
•4  cups so that we pour the tea first and then add the milk,
•4  cups so that we pour the milk first and then add the tea.
•
•We are to choose  4  cups out of the  8  cups.  What is the probability – if one selects the cups
randomly – that we choose the 4 cups where the tea was first correctly?  (I.e., we correctly
recognize whether the tea was first in the cup?)
•  —  By the way, the Lady recognized the cups correctly.

Hypergeometric distribution



Hypergeometric distribution



Hypergeometric distribution



The binomial coefficient in Excel