Statistical Computing This document discusses various probability distributions that are important in data analytics. It begins by defining a probability distribution and giving examples of discrete probability distributions like the binomial distribution. It then discusses properties of discrete and continuous probability distributions. The document also covers specific continuous distributions like the normal, uniform, and Poisson distributions. It provides examples of calculating probabilities and distribution parameters for each type of distribution. In summary, the document presents an overview of key probability distributions and their applications in data analytics and statistics.

1. Statistical Computing
M.V. Padmavati
BIT, Durg
Venkata Padmavathi Metta

2. Probability Distribution
2
A probability distribution is a definition of probabilities of the values of random
variable.
Definition 4.2: Probability distribution
Example 4.3: Given that 0.2 is the probability that a person (in the ages between
17 and 35) has had childhood measles. Then the probability distribution is given
by
X Probability
0 0.64
1 0.32
2 0.04
?
Probability of a person getting measles is 0.2. If a couple is considered, what is the
probability distribution

3. • In data analytics, the probability distribution is important with which
many statistics making inferences about population can be derived .
» In general, a probability distribution function takes the following form
Example: Probability of a person getting measles is 0.2. If a couple is considered,
what is the probability distribution.
3
𝒙 0 1 2
𝑓 𝑥 0.64 0.32 0.04
Probability Distribution
0.64
0.32
0.04
x
f(x)

4. Given a random variable X in an experiment, we have denoted 𝑓 𝑥 = 𝑃 𝑋 = 𝑥 , the
probability that 𝑋 = 𝑥. For discrete events 𝑓 𝑥 = 0 for all values of 𝑥 except 𝑥 =
0, 1, 2, … . .
Properties of discrete probability distribution
1. 0 ≤ 𝑓(𝑥) ≤ 1
2. 𝑓 𝑥 = 1
3. 𝜇 = 𝑥. 𝑓(𝑥) [ is the mean ]
4. 𝜎2
= 𝑥 − 𝜇 2
. 𝑓(𝑥) [ is the variance ]
In 2, 3 𝑎𝑛𝑑 4, summation is extended for all possible discrete values of 𝑥.
Note: For discrete uniform distribution, 𝑓 𝑥 =
1
𝑛
with 𝑥 = 1, 2, … … , 𝑛
𝜇 =
1
𝑛
𝑖=1
𝑛
𝑥𝑖
and 𝜎 2
=
1
𝑛 𝑖=1
𝑛
(𝑥𝑖−𝜇)2
4
Descriptive measures

5. 5
Discrete probability distributions
•Binomial distribution
•Poisson distribution
Continuous probability distributions
•Normal distribution
•Standard normal distribution
•Exponential distribution
Taxonomy of Probability Distributions

6. Defining Binomial Distribution
6
The function for computing the probability for the binomial probability
distribution is given by
𝑓 𝑥 =
𝑛!
𝑥! 𝑛 − 𝑥 !
𝑝𝑥
(1 − 𝑝)𝑛−𝑥
for x = 0, 1, 2, …., n
Here, 𝑓 𝑥 = 𝑃 𝑋 = 𝑥 , where 𝑋 denotes “the number of success” and 𝑋 = 𝑥
denotes the number of success in 𝑥 trials.
Definition 4.3: Binomial distribution

7. Binomial Distribution
7
Example 4.6:Measles study
X = having had childhood measles a success
p = 0.2, the probability that a parent had childhood measles
n = 2, here a couple is an experiment and an individual a trial, and the number
of trials is two.
Thus,
𝑃 𝑥 = 0 =
2!
0! 2 − 0 !
(0.2)0(0.8)2−0 = 𝟎. 𝟔𝟒
𝑃 𝑥 = 1 =
2!
1! 2 − 1 !
(0.2)1(0.8)2−1= 𝟎. 𝟑𝟐
𝑃 𝑥 = 2 =
2!
2! 2 − 2 !
(0.2)2
(0.8)2−2
= 𝟎. 𝟎𝟒

8. Binomial Distributions
• The binomial distribution is
• The mean is
• The standard deviation is
!
( ; , ) (1 ) .
!( )!
x n x
B
n
P x n p p p
x n x

 

.
np
 
(1 )
np p
  
Use for yes/no statistics

9. Example : Binomial Distribution
• What is the probability of guessing correctly at
least six out of ten questions in true-false
objective test.

10. Example : Mean and SD (Binomial Distribution)
• If the probability of defective bolt be 1/10 and
find the following for the binomial distribution
of defective bolts in a total of 400 bolts.
(i) Probability that 40 defective bolts out of 400
(ii) Mean
(iii) Standard deviation

11. Example : Mean and SD (Binomial Distribution)
• Determine binomial distribution for which the
mean is 4 and standard deviation is sqrt(3).

12. Example : Mean and SD (Binomial Distribution)
• Determine n,p and q for binomial distribution
for which the mean is 20 and standard
deviation is 4

13. The Poisson Distribution
There are some experiments, which involve the occurring of the number of
outcomes during a given time interval (or in a region of space).
Such a process is called Poisson process.
Example:
Number of clients visiting a ticket selling counter in a metro station.
13

14. The Poisson Distribution
Poisson distribution is a discreteExample:
Number of clients visiting a ticket selling counter in a metro station.
14

15. Poisson distribution is used to model rare occurrences that occur on
average at rate λ per time interval. Can think of “rare” occurrence in
terms of p  0 and n  ∞. Take these limits so that μ = np.
The Poisson Distribution

16. Poisson Distribution- Definition
Poisson distribution is a discrete probability
distribution and is defined by the probability
function
( ; ) .
!
x
P
P x e
x


 

(x=0,1,2,3……..)
Where μ is a positive constant called the parameter of the
distribution and e= 2.71828 approx.

17. Conditions for Poisson Distribution
• The number of trails(n) is large
• The probability of success (p) is very small
(very close to zero)
• (mean) μ = np is finite

18. Properties of Poisson Distribution
• Poisson distribution is a discrete probability
distribution in which the random variable X
assumes infinite set of values 0,1,2,3,……….
• Poisson distribution is viewed as a limit of the
binomial distribution when n  and p0.
That means μ = np is finite.

19. • The Poisson distribution is
• The mean is
• The standard deviation is
( ; ) .
!
x
P
P x e
x


 

.
x 

.
 

Use for counting statistics
Properties of Poisson Distribution

20. Binomial & Poisson Distributions
• The binomial distribution is
• The mean is
• The standard deviation is
• The Poisson distribution is
• The mean is
• The standard deviation is
!
( ; , ) (1 ) .
!( )!
x n x
B
n
P x n p p p
x n x

 

.
np
 
(1 )
np p
  
( ; ) .
!
x
P
P x e
x


 

.
x 

.
 

Use for yes/no statistics
Use for counting statistics

21. 21
• A random variable x follows Poisson distribution with
parameter 4. Find the probabilities that x assumes the
values [e^-4=0.0183]
(i) 0,1,2,4

22. If 5% of the electric bulbs manufactured by a company
are defective, use Poisson distribution to find the
probability that in a sample of 100 bulbs
(i) None is defective
(ii) 5 bulbs will be defective(given e^-5=0.007)

23. Between the hours of 2 and 4p.m. the average number of
phone calls per minute coming to the switch board of a
company is 2.5. Find the probability that during one
particular minute there will be no phone call at all.
(given e^-2=0.13523 and e^-0.5=0.60650)

24. If a random variable x follows a Poisson distribution
such that P(X=1)=P(X=2), find mean and variance.
Find also probability P(X=0)

25. Continuous Probability
Distributions
CS 40003: Data Analytics 25

26. CS 40003: Data Analytics 26
Continuous Probability Distributions
X=x
f(x)
x1 x2 x3 x4
Discrete Probability distribution
X=x
f(x)
Continuous Probability Distribution

27. • When the random variable of interest can take any
value in an interval, it is called continuous random
variable.
– Every continuous random variable has an infinite,
uncountable number of possible values (i.e., any
value in an interval)
• Consequently, continuous random variable differs
from discrete random variable.
CS 40003: Data Analytics 27
Continuous Probability Distributions

28. Properties of Probability Density Function
The function 𝑓(𝑥) is a probability density function for the continuous random
variable 𝑋, defined over the set of real numbers 𝑅, if
1. 𝑓 𝑥 ≥ 0, for all 𝑥 ∈ 𝑅
2. −∝
∝
𝑓 𝑥 𝑑𝑥 = 1
3. 𝑃 𝑎 ≤ 𝑋 ≤ 𝑏 = 𝑎
𝑏
𝑓(𝑥) 𝑑𝑥
4. 𝜇 = −∝
∝
𝑥𝑓(𝑥) 𝑑𝑥
5. 𝜎 2 = −∝
∝
𝑥 − 𝜇 2 𝑓 𝑥 𝑑𝑥
CS 40003: Data Analytics 28
X=x
f(x)
a b

29. • Find the mean and variance of the p.d.f given
by f(x)=12 x2(1-x) for 0<=x<=1

30. Continuous Uniform Distribution
CS 40003: Data Analytics 30
The density function of the continuous uniform random variable 𝑋 on the
interval [𝐴, 𝐵] is:
𝑓 𝑥: 𝐴, 𝐵 =
1
𝐵 − 𝐴
𝐴 ≤ 𝑥 ≤ 𝐵
0 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Definition 4.8: Continuous Uniform Distribution
• One of the simplest continuous distribution in all of statistics is the
continuous uniform distribution.

31. Note:
a) ∞
−∞
𝑓 𝑥 𝑑𝑥 =
1
𝐵−𝐴
× (𝐵 − 𝐴) = 1
b) 𝑃(𝑐 < 𝑥 < 𝑑)=
𝑑−𝑐
𝐵−𝐴
where both 𝑐 and 𝑑 are in the interval (A,B)
c) 𝜇 =
𝐴+𝐵
2
d) 𝜎2
=
(𝐵−𝐴)2
12
CS 40003: Data Analytics 31
Continuous Uniform Distribution
c
A B
f(x)
X=x

32. week 5 32
The Uniform distribution
(a) X has a uniform[0,1] distribution. The pdf of X is given by:
(b) Uniform[a, b] distribution, b > a. The pdf of X is given by:

33. • The most often used continuous probability distribution is the normal
distribution; it is also known as Gaussian distribution.
• Its graph called the normal curve is the bell-shaped curve.
• Such a curve approximately describes many phenomenon occur in nature,
industry and research.
– Physical measurement in areas such as meteorological experiments, rainfall
studies and measurement of manufacturing parts are often more than
adequately explained with normal distribution.
• A continuous random variable X having the bell-shaped distribution is
called a normal random variable.
CS 40003: Data Analytics 33
Normal Distribution

34. CS 40003: Data Analytics 34
The density of the normal variable 𝑥with mean 𝜇 and variance 𝜎2
is
𝑓 𝑥 =
1
𝜎 2𝜋
𝑒
−
(𝑥−𝜇)2
2𝜎2
− ∞ < 𝑥 < ∞
where 𝜋 = 3.14159 … and 𝑒 = 2.71828 … . ., the Naperian constant
Definition 4.9: Normal distribution
Normal Distribution
• The mathematical equation for the probability distribution of the normal variable
depends upon the two parameters 𝜇 and 𝜎, its mean and standard deviation.
f(x)
𝜇
𝜎
x

35. σ2
µ1
σ1
µ2
Normal curves with µ1<µ2 and σ1<σ2
CS 40003: Data Analytics 35
Normal Distribution
µ1 µ2
σ1 = σ2
µ1 µ2
Normal curves with µ1< µ2 and σ1 = σ2
σ1
σ2
µ1 = µ2
Normal curves with µ1 = µ2 and σ1< σ2

36. Properties of Normal Distribution
– The curve is symmetric about a vertical axis through the mean 𝜇.
– The random variable 𝑥 can take any value from −∞ 𝑡𝑜 ∞.
– The most frequently used descriptive parameter s define the curve itself.
– The mode, which is the point on the horizontal axis where the curve is a maximum
occurs at 𝑥 = 𝜇.
– The total area under the curve and above the horizontal axis is equal to 1.
−∞
∞
𝑓 𝑥 𝑑𝑥 =
1
𝜎 2𝜋
−∞
∞
𝑒
−
1
2𝜎2(𝑥−𝜇)2
𝑑𝑥 = 1
– 𝜇 = −∞
∞
𝑥. 𝑓 𝑥 𝑑𝑥 =
1
𝜎 2𝜋 −∞
∞
𝑥. 𝑒
−
1
2𝜎2(𝑥−𝜇)2
𝑑𝑥
– 𝜎2
=
1
𝜎 2𝜋 −∞
∞
(𝑥 − 𝜇)2
. 𝑒−
1
2
[(𝑥−𝜇)
𝜎2]
𝑑𝑥
– 𝑃 𝑥1 < 𝑥 < 𝑥2 =
1
𝜎 2𝜋 𝑥1
𝑥2
𝑒
−
1
2𝜎2(𝑥−𝜇)2
𝑑𝑥
denotes the probability of x in the interval (𝑥1, 𝑥2).
CS 40003: Data Analytics 36
𝜇 x1 x2

37. • The normal distribution has computational complexity to calculate 𝑃 𝑥1 < 𝑥 < 𝑥2
for any two (𝑥1, 𝑥2) and given 𝜇 and 𝜎
• To avoid this difficulty, the concept of 𝑧-transformation is followed.
• X: Normal distribution with mean 𝜇 and variance 𝜎2
.
• Z: Standard normal distribution with mean 𝜇 = 0 and variance 𝜎2
= 1.
• Therefore, if f(x) assumes a value, then the corresponding value of 𝑓(𝑧) is given by
𝑓(𝑥: 𝜇, 𝜎):𝑃 𝑥1 < 𝑥 < 𝑥2 =
1
𝜎 2𝜋 𝑥1
𝑥2
𝑒
−
1
2𝜎2(𝑥−𝜇)2
𝑑𝑥
=
1
𝜎 2𝜋 𝑧1
𝑧2
𝑒−
1
2
𝑧2
𝑑𝑧
= 𝑓(𝑧: 0, 𝜎)
CS 40003: Data Analytics 37
Standard Normal Distribution
z =
𝑥−𝜇
𝜎
[Z-transformation]

38. CS 40003: Data Analytics 38
Standard Normal Distribution
The distribution of a normal random variable with mean 0 and variance 1 is called
a standard normal distribution.
Definition 4.10: Standard normal distribution
3
2
1
0
-1
-2
-3
0.4
0.3
0.2
0.1
0.0
σ=1
25
20
15
10
5
0
-5
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
σ
x=µ µ=0
f(x: µ, σ) f(z: 0, 1)

40. Statistical Computing
Unit-1
Assignment Questions
1. The overall percentage of failures in a certain examination is 40.
What is the probability that out of group of 6 candidates at least 4
passed the examination? (Use binomial distribution, ans :
1701/3125)
2. Write a short note on Poisson Probability Distribution. The
probability that a man aged 45 years will die within a year is 0.012.
What is the probability that out of 10 such men at least 9 will reach
their forty-sixth birthday [Given e-0.12 = 0.88692]
3. A bag contains 8 red balls and 5 white balls. Two successive draws
of 3 balls are made without replacement. Find the probability that
the first drawing will give 3 white balls and the second 3 red balls.
4. State and prove Bayes’ Theorem of Conditional probability.
5. A man is known to speak truth 2 out of 3 times. He throws a die
and reports that the number obtained is a four. Find the probability
that the number obtained is actually a four.