The document explains three fundamental probability distributions: binomial, Poisson, and normal distribution. Binomial distribution features two possible outcomes in fixed trials, while Poisson distribution addresses independent events over a fixed interval, and normal distribution describes continuous data with a bell-shaped curve. It also covers mean, variance, and their applications in real-life scenarios, alongside comparisons of the distributions.

1. BINOMIAL ,POISSON AND
NORMAL DISTRIBUTION ..
S. SIVA RANJANI

2. BINOMIAL DISTRIBUTION
In probability theory and statistics, the binomial distribution is the discrete probability distribution gives only
two possible results in an experiment, either Success or Failure. For example, if we toss a coin, there could be
only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass
or fail. This distribution is also called a binomial probability distribution.
Number of trials (n) is a fixed number.
The outcome of a given trial is either success or failure.
The probability of success (p) remains constant from trial to trial which means an experiment is conducted
under homogeneous conditions.
The trials are independent which means the outcome of previous trial does not affect the outcome of the next
trial.

3. Binomial Probability Distribution
In binomial probability distribution, the number of ‘Success’ in a sequence of n
experiments, where each time a question is asked for yes-no, then the valued
outcome is represented either with success/yes/true/one (probability p) or
failure/no/false/zero (probability q = 1 − p). A single success/failure test is also
called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a
Bernoulli process. For n = 1, i.e. a single experiment, the binomial distribution is a
Bernoulli distribution.

4. There are two parameters n and p used here in a binomial distribution. The
variable ‘n’ states the number of times the experiment runs and the variable
‘p’ tells the probability of any one outcome. Suppose a die is thrown
randomly 10 times, then the probability of getting 2 for anyone throw is ⅙.
When you throw the dice 10 times, you have a binomial distribution of n =
10 and p = ⅙.

5. FORMULA

6. The binomial distribution formula is for any random variable X, given by;
P(x:n,p) = nCx px (1-p)n-x
Where,
n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of Success in a single experiment
q = Probability of Failure in a single experiment = 1 – p
The binomial distribution formula can also be written in the form of n-Bernoulli trials,
where nCx = n!/x!(n-x)!. Hence,
P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x

7. Binomial Distribution Mean and Variance
For a binomial distribution, the mean, variance and standard deviation for the given number of success
are represented using the formulas
Mean, μ = np
Variance, σ2 = npq
Standard Deviation σ= √(npq)
Where p is the probability of success
q is the probability of failure, where q = 1-p

8. PROPERTIES OF BINOMIAL DISTRIBUTION
The properties of the binomial distribution are:
 There are two possible outcomes: true or false, success or failure, yes or no.
 There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
 The probability of success or failure remains the same for each trial.
 Only the number of success is calculated out of n independent trials.
 Every trial is an independent trial, which means the outcome of one trial does not
affect the outcome of another trial.

9. It is a p.m.f.(probability mass function).
The parameters of Binomial Distribution are ’n’ and ‘p’ where, n - number of trials and p - probability of
success.
Mean of Binomial Distribution = np
Variance of Binomial Distribution = npq (here q=1-p)
Mode of Binomial Distribution is integral part of (n+1)p, if (n+1)p is not an integer. But, if (n+1)p is an
integer, then the distribution has two modal values, (n+1)p and [(n+1)p] - 1.
q- probability of failure, q < 1, hence, npq < np. Thus, Variance is less than Mean.
If p=q=1/2, then the distribution is symmetric about median and if p is not equal to q, then it is skewed
distribution.
Additive property of Binomial Distribution : If X and Y are independent variables such that X follows
Binomial Distribution with (n1, p) and Y follows Binomial Distribution with (n2, p), then (X+Y) follows
Binomial Distribution with (n1+n2, p).
If X1, X2,…., Xn are independent and identically distributed Bernoulli variates each with parameter p, then
their addition follows Binomial Distribution with (n, p).

10. What are the criteria for the binomial
distribution
• The number of trials should be fixed.
• Each trial should be independent.
• The probability of success is exactly the same from one trial to
the other trial.

11. Examples of binomial distribution problems:
The number of defective/non-defective products in a production run.
Yes/No Survey (such as asking 150 people if they watch ABC news).
Vote counts for a candidate in an election.
The number of successful sales calls.
The number of male/female workers in a company
So, as we have the basis let’s see some binominal distribution examples, problems, and solutions from
real life.

12. Binomial Distribution Vs Normal Distribution
• Binomial Distribution Vs Normal Distribution
• The main difference between the binomial distribution and the
normal distribution is that binomial distribution is discrete,
whereas the normal distribution is continuous. It means that the
binomial distribution has a finite amount of events, whereas the
normal distribution has an infinite number of events. In case, if
the sample size for the binomial distribution is very large, then
the distribution curve for the binomial distribution is similar to the
normal distribution curve
•

13. Poission Distribution

14. Poisson Distribution
• Poisson distribution is a theoretical discrete probability and is also known
as the Poisson distribution probability mass function. It is used to find the
probability of an independent event that is occurring in a fixed interval of
time and has a constant mean rate. The Poisson distribution probability
mass function can also be used in other fixed intervals such as volume,
area, distance, etc. A Poisson random variable will relatively describe a
phenomenon if there are few successes over many trials. The Poisson
distribution is used as a limiting case of the binomial distribution when the
trials are large indefinitely. If a Poisson distribution models the same
binomial phenomenon, λ is replaced by np. Poisson distribution is named
after the French mathematician Denis Poisson

15. Poisson distribution
• Poisson distribution definition is used to model a discrete
probability of an event where independent events are occurring in
a fixed interval of time and have a known constant mean rate. In
other words, Poisson distribution is used to estimate how many
times an event is likely to occur within the given period of time. λ
is the Poisson rate parameter that indicates the expected value of
the average number of events in the fixed time interval. Poisson
distribution has wide use in the fields of business as well as in
biology

16. Example :
• Let us try and understand this with an example, customer care
center receives 100 calls per hour, 8 hours a day. As we can see
that the calls are independent of each other. The probability of
the number of calls per minute has a Poisson probability
distribution. There can be any number of calls per minute
irrespective of the number of calls received in the previous
minute. Below is the curve of the probabilities for a fixed value of
λ of a function following Poisson distribution:

17. FORMULA
• Poisson distribution formula is used to find the probability of an
event that happens independently, discretely over a fixed time
period, when the mean rate of occurrence is constant over time.
The Poisson distribution formula is applied when there is a large
number of possible outcomes. For a random discrete variable X
that follows the Poisson distribution, and λ is the average rate of
value, then the probability of x is given by:
• f(x) = P(X=x) = (e-λ λx )/x!
•

18. Where
x = 0, 1, 2, 3...
e is the Euler's number(e = 2.718)
λ is an average rate of the expected value and λ = variance, also λ>0
Poisson Distribution Mean and Variance
For Poisson distribution, which has λ as the average rate, for a fixed interval of
time, then the mean of the Poisson distribution and the value of variance will
be the same. So for X following Poisson distribution, we can say that λ is the
mean as well as the variance of the distribution.

19. Hence: E(X) = V(X) = λ
where
E(X) is the expected mean
V(X) is the variance
λ > 0
Properties of Poisson Distribution
The Poisson distribution is applicable in events that have a large number of rare and independent
possible events

20. Important Notes
• The formula for Poisson distribution is f(x) = P(X=x) = (e-λ λx )/x!.
• For the Poisson distribution, λ is always greater than 0.
• For Poisson distribution, the mean and the variance of the
distribution are equal.

21. PROPERTIES
• The events are independent.
• The average number of successes in the given period of time alone can occur.
No two events can occur at the same time.
• The Poisson distribution is limited when the number of trials n is indefinitely
large.
• mean = variance = λ
• np = λ is finite, where λ is constant.
• The standard deviation is always equal to the square root of the mean μ.
• The exact probability that the random variable X with mean μ =a is given by
P(X= a) = μa / a! e -μ
• If the mean is large, then the Poisson distribution is approximately a normal
distribution

22. Poisson distribution table
• Similar to the binomial distribution, we can have a Poisson
distribution table which will help us to quickly find the probability
mass function of an event that follows the Poisson distribution.
The Poisson distribution table shows different values of Poisson
distribution for various values of λ, where λ>0. Here in the table
given below, we can see that, for P(X =0) and λ = 0.5, the value of
the probability mass function is 0.6065 or 60.65%.

24. NORMAL DISTRIBUTION

26. What are the properties of the normal distribution?
• The normal distribution is a continuous probability distribution that is
symmetrical on both sides of the mean, so the right side of the centre is
a mirror image of the left side.
• The area under the normal distribution curve represents probability and
the total area under the curve sums to one.
• Most of the continuous data values in a normal distribution tend to
cluster around the mean, and the further a value is from the mean, the
less likely it is to occur. The tails are asymptotic, which means that they
approach but never quite meet the horizon (i.e. x-axis).
• For a perfectly normal distribution the mean, median and mode will be
the same value, visually represented by the peak of the curve

27. The normal distribution is often called the bell curve because
the graph of its probability density looks like a bell. It is also
known as called Gaussian distribution, after the German
mathematician Carl Gauss who first described it.

28. What is the difference between a normal distribution and a
standard normal distribution?
A normal distribution is determined by two parameters the mean and the variance. A normal distribution with a
mean of 0 and a standard deviation of 1 is called a standard normal distribution.

29. Why is the normal distribution important?
 The bell-shaped curve is a common feature of nature and psychology
The normal distribution is the most important probability distribution
in statistics because many continuous data in nature and psychology displays
this bell-shaped curve when compiled and graphed.
For example, if we randomly sampled 100 individuals we would expect
to see a normal distribution frequency curve for many continuous variables,
such as IQ, height, weight and blood pressure.

30.  Parametric significance tests require a normal distribution of the samples' data points
The most powerful (parametric) statistical tests used by psychologists require
data to be normally distributed. If the data does not resemble a bell curve researchers may
have to use a less powerful type of statistical test, called non-parametric statistics.
 Converting the raw scores of a normal distribution to z-scores
We can standardized the values (raw scores) of a normal distribution by
converting them into z-scores.
This procedure allows researchers to determine the proportion of the values
that fall within a specified number of standard deviations from the mean (i.e. calculate the
empirical rule).

31. What is the empirical rule formula?
• The empirical rule in statistics allows researchers to determine
the proportion of values that fall within certain distances from the
mean. The empirical rule is often referred to as the three-sigma
rule or the 68-95-99.7 rule.

33.  If the data values in a normal distribution are converted to standard score (z-score)
in a standard normal distribution the empirical rule describes the percentage of the
data that fall within specific numbers of standard deviations (σ) from the mean (μ)
for bell-shaped curves.
 The empirical rule allows researchers to calculate the probability of randomly
obtaining a score from a normal distribution.
 68% of data falls within the first standard deviation from the mean. This means there
is a 68% probability of randomly selecting a score between -1 and +1 standard
deviations from the mean.

34.  95% of the values fall within two standard deviations from the
mean. This means there is a 95% probability of randomly selecting
a score between -2 and +2 standard deviations from the mean.
 99.7% of data will fall within three standard deviations from the
mean. This means there is a 99.7% probability of randomly
selecting a score between -3 and +3 standard deviations from the
mean.