Download Some Important Discrete Distributions Binomial Distribution

Some Important Discrete Distributions
Binomial Distribution
The binomial distribution is the discrete probability distribution of the number of successes in a
sequence of n independent experiments, each of which yields success with probability p and failure
with probability
1
. Such a success/failure experiment is also called a Bernoulli experiment or
Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution. The binomial
distribution is the basis for the popular binomial test of statistical significance. The binomial
distribution is frequently used to model the number of successes in a sample of size n drawn with
replacement from a population of size N.
In general, if the random variable X follows the binomial distribution with parameters n and p,
we write X ~ BIN(n, p). The probability of getting exactly
successes in
trials is given by the
probability mass function:
,
where
= total number of trials ,
1
, ,…,
= probability of success
= probability of failure,
= number of success in n trials
This is a discrete probability distribution and it should be noted that
...
The requirements for using the binomial distribution are as follows:
•
There are only two possible outcomes in each trail.
•
All trials have the same probability for a particular outcome in a single trial.
•
That is, the probability in a subsequent trial is independent of the outcome of a previous
trial. Let this constant probability for a single trial be p.
•
The number of trials, n, must be fixed, regardless of the outcome of each trial.
Example 1
On the basis of past experience, the probability that a certain electrical component will be
satisfactory is 0.98. In a sample of five components, what is the probability of finding
(a) zero,
(b) exactly one,
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(c) exactly two,
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(d) two or more defective component?
Dr. Mohamed Hussein
The m.g.f of the binomial distribution is given by
Proof
The mean and variance of the binomial distribution are
and
Proof
′
′
′′
0
1
′′
0
1
1
1
The standard deviation is always given by the square root of the corresponding variance, so the
standard deviation for the binomial distribution is
Example 2
A company is considering drilling four oil wells. The probability of success for each well is 0.40,
independent of the results for any other well. The cost of each well is $200,000. Each successful
well achieves profit of $600,000.
a) What is the probability that one or more wells will be successful?
b) What is the expected number of successes?
c) What is the expected profit?
BE208 (Fall 2012)
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Dr. Mohamed Hussein
Poisson Distribution
A discrete random variable
0, ~
, if for
is said to have a Poisson distribution with parameter
0, 1, 2, . .. the probability mass function of
!
,
is given by:
0,1, …
Notice that:
!
1
!
The Poisson distribution expresses the probability of a given number of events occurring
in a fixed interval of time and/or space if these events occur with a known average rate and
independently of the time since the last event. (The Poisson distribution can also be used for the
number of events in other specified intervals such as distance, area or volume.)
The average number of collisions occurring in a week during the summer months at a particular
intersection is 2.00. Assume that the requirements of the Poisson distribution are satisfied.
a) What is the probability of no collisions in any particular week?
b) What is the probability that there will be exactly one collision in a week?
c) What is the probability of exactly two collisions in a week?
d) What is the probability of finding not more than two collisions in a week?
e) What is the probability of finding more than two collisions in a week?
The Poisson distribution can be used as an approximation of the binomial distribution if n is
sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson
distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller
than or equal to 0.05, and an excellent approximation if
100 and
10.
Example 3
Suppose that 1% of all transistors produced by a certain company are defective. A new
model of computer requires 100 of these transistors, and 100 are selected at random from the
company's assembly line. Find the probability of obtaining 3 defectives.
Solution
The exact probability of obtaining 3 defectives using the binomial distribution with
and
.01
100 is
BE208 (Fall 2012)
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Dr. Mohamed Hussein
.
.
.
1, is
While the Poisson approximation with
.
!
The m.g.f of the Poisson distribution is given by
Proof
!
!
The mean and variance of the Poisson distribution are
and
Proof
′
.
′
′′
.
0
.
.
′′
0
Negative Binomial Distribution
Suppose there is a sequence of independent trials, each trial having two potential outcomes
called “success” and “failure”. In each trial the probability of success is p and of failure is (1 − p).
We are observing this sequence until a predefined number k of successes has occurred. Then the
random variable X is the total number of trials needed to get k successes will have the negative
binomial (or Pascal) distribution:
,
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,
,…
Dr. Mohamed Hussein
The m.g.f of the negative binomial distribution is given by
1
The mean and variance of the negative binomial distribution are
and
Geometric Distribution
The geometric distribution is also related to a sequence of Bernoulli trials. The random
variable X of interest is defined to be the number of trails required to achieve the first success.
The probability mass function of
is given by
,
, ,…
The m.g.f of the geometric distribution is given by
1
The mean and variance of the geometric distribution are
and
Example 4
A certain experiment is to be performed until a successful result is obtained. The trails are
independent and the cost of performing the experiment is $25,000; however, if a failure results, it
costs $5000 to “set up” the next trail. The experimenter would like to determine the expected cost of
the project if the probability of success on a single trail is 0.25, and the cost function is given by
25000
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5000
1 .
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Dr. Mohamed Hussein