This document provides an overview of probability distributions including binomial, Poisson, and normal distributions. It discusses key concepts such as: - Binomial distributions describe experiments with two possible outcomes and fixed number of trials. - Poisson distributions model rare events with sample sizes so large one outcome is much more common. - Normal distributions produce bell-shaped curves defined by the mean and standard deviation. They are widely used in statistics.

1. Lecture 5
Probability Distribution
Dr. Ashish. C. Patel
Assistant Professor,
Dept. of Animal Genetics & Breeding,
Veterinary College, Anand
STAT-531
Data Analysis using Statistical Packages

3. • We have discussed about frequency distributions
that are based on the actual observations.
• There are some other distributions that are not
obtained by actual observations or experiments
but are assumed mathematically on the basis of
certain assumptions.
• These distributions are known as Probability
distributions or Mathematical distributions or
Theoretical Distributions.
• These distributions are based on the laws of
probability.

4. • Three main probability distributions:
binomial, Poisson and normal distributions.
• Some other distributions, such as t-
distribution and chi-square distributions also

5. BINOMIAL DISTRIBUTION
• This distribution was discovered by Swiss
Mathematician James Bernoulli hence it is also known
as Bernoulli’s distribution.
• There are many trials/study/experiments that have
only two outcomes.
• For example, birth of a male or a female calf,
occurrence of head or tail in tossing a coin, dead or
alive, success or failure, presence or absence of a
particular characteristic, etc.
• Such single trials are often called binomial or
Bernoulli’s trials.
• The binomial distribution can be defined as the
probability distribution of discrete variable where
there are only two possible out comes for each trial of
an experiment.

6. • For example, let’s suppose you wanted to know
the probability of getting a 1 on a dice roll.
• if you were to roll a dice 20 times, the
probability of rolling a one on any throw is 1/6.
• Roll twenty times and you have a binomial
distribution of (n=20, p=1/6).
• If the probability of the dice landing on an even
number, the binomial distribution would then
become (n=20, p=1/2).

7. • Binomial distributions must also meet the
following three criteria:
1. The number of observations or trials is fixed. If
we toss a coin once, the probability of getting a
tails is 50%. If we toss a coin a 20 times, the
probability of getting a tails is very, very close to
100%.
2. Each observation or trial is independent. In
other words, none of our trials have an effect on
the probability of the next trial.
3. The probability of success (tails, heads, fail or
pass) is exactly the same from one trial to
another.

8. • The binomial distribution formula is:
b(x; n, P) = nCx * Px * (1 – P)n – x =
• Where:
b = binomial probability
x = total number of “successes” (pass or fail, heads or
tails etc.)
P = probability of a success on an individual trial
n = number of trials

9. Properties (characteristics) of binomial distribution
• The sum of probabilities corresponding to 0,1,2,….,n successes is
unity.
• The mean of binomial distribution is np and its variance
is npq.
• The quantity ‘n’ and ‘p’ are the parameters determine
the shape and location of distribution.
• If p = q = ½, the distribution obtained is symmetrical
• A binomial distribution is positively skewed if p < ½ and
it is negatively skewed if p > ½.
• The variance of a binomial distribution is always less
than its mean
• p and q are always less than one but not equal to zero.

10. Suppose 10 MCQs are there in Examination and Each MCQ has 4
options A, B, C, D
• 1 A B C D
• 2 A B C D
• 3 A B C D
…...
• 10 A B C D
• Suppose student select answer randomly
• What is probability all answer incorrect?
• Here n= 10, probability of correct answer = ¼ = p
• P(0) =
• P(0) = 0.056
• P(1) =?

11. POISSON DISTRIBUTION
• In the binomial distribution, we have relatively small
samples in which two alternative states occurred at
varying frequencies.
• however, we study cases in which sample size n is very
large, and one of the events (represented by probability
q) is very much more frequent than the other
(represented by p).
• The expansion of the binomial term (p + q)n is quite
tiresome when n is large. For example, if we want to
expand the expression (0.001 + 0.999)1000.
• In such cases, we are generally interested in only one
tail of the distribution. It is customary in such cases to
compute another distribution called Poisson
distribution discovered by French Mathematician
Simcom Denis Poisson.

12. • The Poisson distribution is a discrete probaility
distribution of the number of times a rare event
occurs.
• The number of times that an event does not occur
is infinitely large. For example,
• birth of a calf having congenital defect,
• presence of an abnormal cyst in an internal organ,
• persons killed by kicking of horses,
• number of people dying in road accidents, etc.
• In all these examples the probability of happening
of the given events is very low as compared to their
complements whose numbers are infinitely large.

13. • The Poisson Distribution formula is:
P(x; μ) = (e-μ) (μx) / x!
The average number of major storms in your city is 2 per
year. What is the probability that exactly 3 storms will
hit your city next year?
μ = 2 (average number of storms per year, historically)
x = 3 (the number of storms we think might hit next year)
e = 2.71828 (e is Euler’s number, a constant)
Now, P(x; μ) = (e-μ) (μx) / x!
= (2.71828 – 2) (23) / 3!
= (0.13534) (8) / 6
= 0.180
• The probability of 3 storms happening next year is
0.180, or 18%

14. Properties of Poisson distribution
• The mean of this distribution in np.
• The variance of Poisson distribution is also equal to
its mean, i.e. np. This would be expected, since the
binomial expression, npq, tends to np when q tends
to 1.
• The parametric mean m is the constant of the
Poisson distribution.

15. • Suppose 100 pages of book are randomly
selected. What is probability that there
are no typos (typing mistakes/ error)?
• X = 0 and μ = 1.2
• P (0) = 0.30

16. NORMAL DISTRIBUTION
• The normal distribution was discovered by Abraham
de Moivre. Later on Carl Friedrich Gauss gave precise
formulation for calculation of Normal Distribution so, it
is also known as Gaussian distribution
• Normal distribution is a distribution of continuous
variable.
• Unlike discrete probability distributions, we cannot
evaluate the probability of the variable exactly equal to
a given value such as 3 or 3.5 in a continuous
distribution.
• We can only estimate the frequency of the
observations falling between two limits.
• This is so because the area of the curve corresponding
to any point along the curve is an infinitesimal.

17. Properties of the Normal Distribution
• The normal probability density function can be represented
by the expression.
Z indicates the height of the ordinate of the curve, which
represents the density of the items.
• The curve is bell shaped, unimodal (has only one peak) and
symmetrical around the mean Therefore the mean, median
and mode of a normally distributed curve are all at the same
point, i.e., mean = median = mode. The tails of the curve
taper towards the base line on both the sides asymptotically,
i.e., they do not touch the base line.
• A normal distribution is defined by a mean (average) of zero
and a standard deviation of 1.0, with a skew of zero and
kurtosis = 3.

19. • The following percentage of items in a normal
frequency distribution lie within the indicated
limits:
• μ ± σ contains 68.26 % of the items
• μ ± 2σ contains 95.45 % of the items
• μ ± 3σ contains 99.73 % of the items
• Conversely,
• 50 % of the items fall between μ ± 0.674 σ
• 95 % of the items fall between μ ± 1.960 σ
• 99 % of the items fall between μ ± 2.576 σ

20. • The coefficient of skewness in a normal curve is zero while
the coefficient of kurtosis is 3.
• A normal curve with zero mean and unit variance (or
standard deviation) is referred to as the standard normal
curve.
• The first and the third quartiles are equidistant from the
median. Thus: (Q3 – Median) = (Median – Q1)
• The height of a normal distribution curve is maximum at its
mean. It means that the mean ordinate divides the curve
into two equal parts.
• The points of inflexion (the points at which the curve
changes its direction) is each one s away from the mean
ordinate.
• Various ordinates at different distances of s from the mean
ordinate stand in a fixed ratio to the height of the mean
ordinate. For example, the height of the ordinate at 1 s
distance on either side of the mean ordinate is 60.65 % of
the height of the mean ordinate.

21. Applications of Normal Distribution
• The normal distribution is most widely used theoretical distribution
because it possesses a lot of mathematical properties.
• It is extremely used in a wide variety of fields like physical, natural
and social sciences for making various types of analyses.
• It forms the basis of tests of significance because of the fundamental
assumption that the population from which samples are drawn
should be normally distributed.
• It is widely used in planning of experiments. This is so because unless
the assumption of normality is satisfied, valid inferences can not be
made from the results.
• It is highly useful in statistical quality control in industries for setting
up of the control limits.
• It is most often used in the large sampling theory to find the
estimates of the parameters and confidence limits from statistic.