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Topic
-Ve Binomial Distribution & Geometric
Distribution.
Class
BS(CS) 2nd Semester
Submitted To
M.Yahya
Submitted From
M. Zishan Zafar, M. Waqas.
Submission Date
17-08-2005
What is Distribution
Statistical arrangement of values of a variable showing
their observed or theoretical frequency of occurance.
What is binomial distribution
In probability, a binomial distribution gives the probabilities of k
outcomes A (or nk outcomes B) in n independent trials for a two-outcome experiment in
which the possible outcomes are denoted A and B.
(OR)
The theoretical frequency distribution of events that have two
possible outcomes which are “success” or “failure”
Explaination
A probability distribution that applies to experiments involving sequences of
independent trials in which only two possible outcomes (eg, success or failure) can result on
each trial. If p is the probability of success on each trial, and q = 1 - p the probability of
failure, then the probability of success occurring x times in n trials is given by the binomial
distribution
Negative Binomial Distribution
Definition:
If repeated independent trials can result in a success with probability p and a
failure with probability q = 1-p, then the probability description of the random variable X the
number of trials on which success occurs is given by
b*(x; k, p) =
pk qx-k,
x = k, k+1, k+2, …
Explanation
This distribution is also called the Pascal distribution after the French
mathematician Blaise Pascal (1623-1662) The distribution is found to occur in many
biological situations and in inverse sampling from a binomial population.
In binomial experiment, the number of successes varies and the number of trials is fixed. But
there are the experiments in which the number of successes fixed and the number of trials
varies to produce the fixed number of successes. Such an experiments are called negative
binomial experiments.
When denote the number of trials to produce k successes in a negative binomial
experiment, it is called a negative binomial variable and its p.d. is called the negative
binomial distribution. When the negative binomial r.v. X assumes a value x on which the kth
success occurs, the negative binomial distribution is given by
b*(x; k, p) =
pk qx-k,
x = k, k+1, k+2, …
Conditions For Negative Binomial Distribution
(1) The out comes of each trial may be classified into one or two categories: success (S)
and failure (F).
(2) The probability of success, denoted by P remains constant for all trials.
(3) The successive trials are all independent.
(4) The experiment is repeated a variable number of times to obtain a fixed number of
successes.
Example:
Find the probability that a person tossing 3 coins will either all heads or all tails
for the second time on the fifth toss.
Solution:
Now here x= 5, k = 2 and p = ¼
b*(x; k, p) =
x-1
k-1
b*(5; 2, ¼) = 4
1
pk qx-k,
1
4
x = k, k+1, k+2, …
3
4
= 4!__ . 33
1! 3!
45
= 37
256
Geometric Distribution
Definition
If repeated independent trials can result in a success with probability p and a
failure with probability q = 1-p then the probability distribution of the random variable X, the
number of the trial on which the first success occurs, is given by
g(x; p) = pqx-1
for
x = 1, 2, 3, …
when an experiment consists of independent trials with probability p of
success and the trials are repeated until the first success occurs, it is called a
geometric experiment.
Conditions for Geometric Distribution
(1) The outcomes of each trial may be classified into one of two categories, success and
failure.
(2) The probability of success p remains constant for all trials.
(3) The success trials are all independent.
(4) The experiment is repeated a variable number of times until the first success is
obtained.
If X represents the number of trials needed for the first success, then X is called a geometric
r.v and its p.d. is called the geometric distribution. It has only one parameter p and is
denoted by g(x ; p).The geometric distribution drive its name from the fact that its successive
terms constitute a geometric progression. Since a Geometric r.v represents how long one
has to wait for her success.
Example:
Find the probability that a person flipping a balanced coin requires 4 tosses to
get a head.
Solution:
Here x= 4 and p = ½ using geometric distribution we have
g(4 ; ½) = ½ (1/2)3
= 1/16
Summary
: Now here we discuss two topics which are related to distribution. First we discuss
about –ve binomial distribution a –ve binomial distribution occurs when
(1) The out comes of each trial may be classified into one or two categories: success (S)
and failure (F).
(2) The probability of success, denoted by P remains constant for all trials.
(3) The successive trials are all independent.
(4) The experiment is repeated a variable number of times to obtain a fixed number of
successes.
And similarly a geometrical distribution occurs when
(1) The outcomes of each trial may be classified into one of two categories, success and
failure.
(2) The probability of success p remains constant for all trials.
(3) The success trials are all independent.
(4) The experiment is repeated a variable number of times until the first success is
obtained.
Conclusion
If repeated independent trials can result in a success with probability p and a
failure with probability q = 1-p, then the probability description of the random variable X the
number of trials on which success occurs is given by
b*(x; k, p) =
and
pk qx-k,
x = k, k+1, k+2, …
If repeated independent trials can result in a success with probability p and a
failure with probability q = 1-p then the probability distribution of the random variable X, the
number of the trial on which the first success occurs, is given by
g(x; p) = pqx-1
for
x = 1, 2, 3, …