probability distribution

1.  Carl Friedrich Gauss.
 discover
 Jacob Bernoulli
 Binomial distribution
 Sir Framin Galton
 Pioneered
 Normal distribution
Siméon Denis
Poisson

2.  it was discovered by Carl Friedrich Gauss.
 Pioneered by Sir Framin Galton 1822 – 1911
 Who studied the quantitative characters and their inheritance
 A distribution that is not a normal distribution is called as
anormal never called as abnormal distribution.
 Describes the distribution of any observation which deviates
by the chance from mean value.
 Eg. 100 lit/day.
 We may have most of value is high and low then mean.
 The graphical curve is bell in shaped.
 The curve is also known as normal distribution curve or
Gaussian curve.
 The distribution is variously called as error law, normal law,
Gaussian law

3. Normal Distribution
 Applied to single variable continuous data
e.g. heights of plants, weights of lambs, lengths of
time
 Used to calculate the probability of occurrences less
than, more than, between given values
e.g. “the probability that the plants will be less than
70mm”,
“the probability that the lambs will be heavier than
70kg”,
“the probability that the time taken will be between 10
and 12 minutes”

4. The normal destitution is actually characterized by the
following equation.
f (y) =
1
𝝈√𝟐𝝅
𝒆 −
𝒚−𝝁 𝟐
𝟐𝝈𝝅
• f(y) : frequency of particular value of the variable
Y
• 𝜋 =
22
7
• e = 2.7184

5. Normal curve.

7. When the frequency distribution is asymmetrical,
the distribution is known as skewed.
Nature of skewness :
MENE=MEDIAN=MODE
MENE>MEDIAN>MODE MENE<MEDIAN<MODE

8. Binomial Distribution
 Imagine a simple trial with only two possible
outcomes
Success (S) Or Live (L)
Failure (F) Death (D)
 Discover in 18 𝑡ℎ
century by Swiss mathematics
Jacob Bernoulli 1654-1705
 It is also called as Dichotomous classification.
Examples
Toss of a coin (heads or tails)
Sex of a newborn (male or female)
Survival of an organism in a region (live or die)

9.  Expression of binomial theorem is (𝑎 + 𝑏) 𝑛
= 1
 a & b respective probabilities
 n = no. of trials
 The probability of having two male and female
children in a family of four children can
calculated as follows :
I. Male (a) ½
II.female (b) ½
III.Number of child = 4
(𝒂 + 𝒃) 𝒏

10. (𝒂 + 𝒃) 𝟒 =𝒂 𝟒 + 4 𝒂 𝟑 b + 6𝒂 𝟐 𝒃 𝟐 4 𝒂𝒃 𝟑 +b 𝟒
Therefore p= 6𝒂 𝟐 𝒃 𝟐
= 6 ×
𝟏
𝟐
𝟐
×
𝟏
𝟐
𝟐
= 6×
𝟏
𝟒
𝟒
= 6×
𝟏
𝟏𝟔
=
𝟑
𝟖
=0.375
Thus the probability in family of four children having 2 boys
and 2 grits is 3/8 . it means three families out of eight are
expected to have 2 boys and 2girls

11. A simple formula based on factorial (!)
binomial theorem can be used to calculate the
probability by a short-cut method.
P =
𝑛!
𝑠!𝑡!
× 𝑝 𝑠
𝑞 𝑡
Where n=total no of event
S= the number of time (a) occurs
T=the number of time (b) occurs
Therefore n= s+t
Factorial 5! means 5’ 4’ 3’ 2’ 1

12.  What is the probability that a family with five children will
have 3 boy and 2 girls.
The probability of child to be a boy : p = ½
The probability of child to be a girl : p = ½
P =
𝒏!
𝒔!𝒕!
× 𝒑 𝒔 𝒒 𝒕
P =
𝟓!
𝟑!𝟐!
× ½ 𝟑
. ½ 𝟐
=
𝟓×𝟒×𝟑×𝟐×𝟏
𝟑×𝟐×𝟏×𝟐×𝟏
×
𝟏
𝟐
×
𝟏
𝟐
×
𝟏
𝟐
×
𝟏
𝟐
×
𝟏
𝟐
×
𝟏
𝟐
=10×
𝟏
𝟑𝟐
=0.032

13.  Mean for binomial distribution :
𝝁 = 𝒏. 𝒑
n is no of independent trial
P is the probability of success
Standard deviation for binomial distribution
𝝈 = 𝒏. 𝒑. 𝒒
n is no of trial
P is the probability of
success
q is the probability of failure

14. Pascal's triangle
 The expression of binomial theorem
is
(𝒂 + 𝒃) 𝒏 = 1
 For each value of n the binomial
expression can be expanded
 The numerical coefficient preceding
each expression can be determine
using Pascal‘s triangle.
 The coefficients denoted the
number of ways by which a
particular combination of events
can occur

15. Multinomial expression
 The binomial theorem can be expanded to including
more then two events. the multinomial expansion
(p+q+r……) 𝒏
can be represented by the general
formula
P =
𝒏!
𝒔!𝒕!𝒖!
× 𝒑 𝒔
𝒒 𝒕
× 𝒓 𝒖
……..
Here, p+q+r……. =1 and
s+t+u….….=n

16. Poisson Distributions
 French mathematician & physicist Siméon Denis
Poisson describe in first 1837
 Is also discreet frequency distribution like binomial
distribution but with a difference
 It is a distribution of the number of times a rare discrete
event occurs in a continuum of space or time.
 It is may be obtained by a limited cases of binomial
probability distribution under the following condition
1)n the number of trial is indefinitely large
2)p the constant probability of success for each trial is
indefinitely small
3)np = m is finite

17.  Poisson Distributions is generally studied in a spatial (
space such as length, area, volume etc.) (time such as
second, minute, hour, day, month, year etc.)
 Poisson event is a rare event because the probability of its
occurrence is very small.
Example of Poisson event
 Poisson events in a continuum of time
o Number of lightning during a half- hour period of
thunderstone
o Numbers of earthqua1kes per year
 Poisson events in a continuum of time
o Number of RBCs / WBCs seen in one square of a
hemocytometer
o Number of bacterial colonies in a plate culture medium
o Number of printing errors in a page of book

18.  Derivation of Poisson probability distribution :
𝒆−𝒎
× 𝒆 𝒎
= 𝒆−𝒎
(
m 𝟎
𝟎!
+
m 𝟏
𝟏!
+
m 𝟐
𝟐!
+
m 𝟑
𝟑!
+
m 𝟒
𝟒!
)
=
𝒆−𝒎m 𝟎
𝟎!
+
𝒆−𝒎m 𝟏
𝟏!
+
𝒆−𝒎m 𝟐
𝟐!
+
𝒆−𝒎m 𝟑
𝟑!
+
𝒆−𝒎m 𝟒
𝟒!
+ ……..
 mean number of occurrence of the event = m
 Successive term of expression = 𝒆−𝒎
× 𝒆 𝒎
 Time or space = x
 P occurrence of Poisson event x time =
𝒆−𝒎m 𝒙
𝒙!

19.  Properties of Poisson Distributions
 The Poisson Distributions is defined by a single
parameter the mean
m= 𝑓𝑥/ 𝑓𝑥
 Suppose the n which should be a large number and p
the probability of the occurrence of the event which
should he very small value then
m= np
 The variance of Poisson distribution is equal to the
mean of the distribution i.e. variance =m therefore
the S.D. of Poisson distribution= 𝑣𝑎𝑟𝑟𝑖𝑎𝑛𝑐𝑒 = √𝑚

20.  Application of Poisson distribution :
1)It is use to find whether distribution of plant or
animal in there environment is random or has
any tendency for clumping or repulsion.
2)It can also be used to understand whether the
occurrences of incidences of diseases such as
malaria, polio, chicken pox etc are rare or
otherwise