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Suchithra's Statistics Classes -- Negative Binomial Distribution

Negative Binomial Distribution
Or
Pascal Distribution
(Based on complementary Statistics of
Bsc , University of Calicut)

Suchithra's Statistics Classes -- Negative
Binomial Distribution
A binomial experiment is an experiment
consisting of a fixed number of independent
trials each with two possible outcomes,
occurrence or non occurrence
(success and failure), and with the same
probability of occurrence (success).
Or; a binomial experiment is an experiment
consisting of a fixed number of
independent bernoulli trials.

The negative binomial experiment is
almost the same as a binomial
experiment with one difference:
In a binomial experiment we have a fixed
number of trials.
But in the negative binomial experiment
we have fixed number of successes.
we repeats the performing Bernoulli trials,
until the rth success occurs.
i.e; number of trials will vary.

Example.
You flip a coin repeatedly for a fixed
number of times and count the number
of times the coin turns on heads is
binomial.
If you continue flipping the coin until it
has turned a particular number of
heads say 5 times on heads then it is
negative binomial since the number of
trials will vary from person to person.

A negative binomial experiment is a statistical
experiment that has the following properties:
•The experiment consists of x+r repeated
trials, where r is the required number of
successes.
•Each trial can result in just two possible outcomes.
say a success and the other, a failure.
•The probability of success, denoted by p, is the
same on every trial.
•The trials are independent--the outcome on
one trial does not affect the outcome on other
trials.
•The experiment continues until r successes are
observed, where r is specified in advance.

Definition
A random variable X is said to follow a
negative binomial distribution if its
probability mass function is given by

Consider a Bernoulli sequence of trials with
probability of success p and probability of
failure q.
Let f(x) be the probability that (x+r) trials will be
required to produce r successes.
i.e; in (x+r-1) trails we get r-1 successes
and the next [(x+r)th ] is a success.

In a negative binomial distribution

Let Y be the number of trials needed to get r
successes.
Y→b*(y; r, p): denotes negative BD
Here y is x+r of the first form of negative binomial
distribution,who’s pmf is

A random variable X is said to follow
a negative binomial distribution if its probability
mass function is given by
The negative binomial experiment is almost the same as
a binomial experiment with one difference: a binomial experiment
has a fixed number of trials, but in negative binomial experiment
we have fixed number of successes
REWIND

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Negative Binomial Distribution

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Negative Binomial Distribution