There are three probability Distributions: Binomial Poisson Normal

1. Probability Distribution
Md. Saiful Islam
Dept. of Pharmaceutical Sciences
North South University
Facebook Group: Pharmacy Universe

2. There are three probability Distributions:
• Binomial
• Poisson
• Normal
The Bernoulli Process: A sequence of Bernoulli trials forms a Bernoulli
process under the following conditions.
1. Each trial results in one of two possible mutually exclusive outcomes. One
of the possible outcomes is denoted as success, and the other is denoted
as failure.
2. The probability of success denoted by p, remains constant from trial to
trial. The probability of failure is 1-p is denoted by q.
3. The trials are independent that is outcome of any particular trial is not
affected by the outcome of any other trial.

3. Example:
We are interested in being able to compute the probability of x successes in
Bernouli trials. For example in a certain population 52 percent of all recorded
births are males. We interpret this to mean that the probability of a recorded
male birth is 0.52. If we randomly select five birth records from this
population, what is the probability that exactly three of records will be for
male births?
Solution:
It will be convenient to assign number 1 to a success (record for a male birth)
and number 0 to failure (record of female birth)
The process that eventually results in a birth recorded we consider Bernoulli
process.
Suppose the five records selected resulted in this sequence of sexes:
MFMMF
In coded form we write this as
10110
Since the probability of success is denoted by p and the probability of failure
is denoted by q, the probability of the sequence of outcomes is found by
means of the multiplication rule to be
P ( 1,011,0)= pqppq = q2p3

4. When draw a single sample of size five from the population specified, we obtain
only one sequence of successes and failures. The question is what is the
probability of getting sequence number 1 or sequence number 2 ….. or
sequence number 10?
Since in population, p=0.52, 1-q= p = (1-0.52) = 0.48, the answer to the question
is
10( 0.48)2 ( 0.523 ) =0.32
In our example we let x=3, the number of success, so that n-x= 2, the number
of failures. We can write the probability of obtaining exactly x successes in n
trials as the f(x) = ncx px qx-n , for x= 0,1,2 3,…..n
= 0, elsewhere
The expression is called the binomial distribution We f (x) = P (X=x), where X
is the random variable, the number of successes in n trials.

5. Poisson Distribution
The next discrete distribution is the Poisson distribution. The Distribution is
named after French Mathematician Simeon Denis Poisson. If x is the number
of occurrence of some random event, the probability that x will occur is given
by
ι
λ
x
e xx−
∞
f(X=x)=
,for x= 0,1,2,…………..
The Greek letter λ is called the parameter of the distribution and is the average
number of occurrences of the random event in the interval. E is constant and its
value is 2.7138

6. The Poisson Process
We have seen that the Binomial distribution results from a set of
assumptions about an underlying process yielding a set of numerical
observations. Such is also the case with the Poisson distribution.
The following statements describe what is known as the Poisson Process.
1. The occurrences of events are independent
2. Theoretically an infinite number of occurrences of the event must be possible in
the interval
3. The probability of the single occurrence of the event in a given interval is
proportional to the length of the interval
4. In any infinitely small portion of the interval, the probability of more than one
occurrence of the event is negligible.

9. Continuous Probability Distribution
The probability distributions considered thus far, binomial ad Poisson, are
distribution of discrete variables. Let us now consider continuous probability
distribution. We already define a continuous variable is one that can assume any
value within a specified interval of values assumed by the variables. Consequently,
between any two values assumed by a continuous variable, there exist an infinite
number of values.
A nonnegative function f (x) is called a probability distribution (some times also
called probability density function) of the continuous random variable X if the total
area bounded by its curve, the x-axis, and perpendiculars erected at any two points
a and b gives the probability that X is lies between the points a and b
The most important continuous distribution in all of statistics – is the Normal
distribution.