Data Distribution &The Probability Distributions

Data Distribution &
The Probability Distributions
Prepared by:
Maha Sabri Altememe
1st. Babylon International Conference on
Information Technology and Science 2021
Lecture 1

First Day :
Outline
Concepts
&Introduction.
Common Data Types

Introduction
 Data is a collection of facts, such as numbers, words, measurements,
observations or just descriptions of things.
 Data are the basic inputs to any decision making process.
 Proper knowledge of data types is necessary to analyze data sets with
appropriate statistical methods.

Data Distributions :
graphical methods of organizing and display useful
information
•Dot plots :show numerical values plotted on scale
•Histograms :display data in ranges
•Pie Charts: show how much each category represents as a
proportion of the whole, by using a circular format with different-sized
“slices” for different percentages of the whole.
• Bar Graphs: use a series of rectangular bars to show absolute
values or proportions for each of the categories.

A histogram is a graph used to represent the frequency of continuous data.

Symmetric
Skewed Right
Skewed left
Uniform
Shapes of distributions

Data Distribution &
Prepared by:
Maha Sabri Altememe
https://www.mathsisfun.com/data/random-variables-mean-variance.html
Lecture 2

Random Variable
is a function that associates a real number with each element in the sample
space.
We shall use a capital letter, say X, to denote a random variable and its
corresponding small letter, x in this case for one of its values
X=x
For instance, a random variable representing the number of cars sold at a
particular dealership on one day would be discrete, while a random variable
representing the weight of a person in kilograms (or pounds) would be
continuous.

Probability Distribution
is a formula or a table used to assign probabilities to each possible value of a
random variable X. There are descriptive statistics used to explain where the
expected value may end up. Some of which are:
•Mean (expacted): μ = Σxp
•Median
•Standard deviation
μ = Σxp = 0.1+0.2+0.3+0.4+0.5+3 = 4.5

A probability distribution may be either
A discrete distribution means that X can assume one of a countable
(usually finite) number of values, or
Continuous distribution means that X can assume one of an infinite
(uncountable) number of different values.

Cumulative distribution function

Discrete Distribution Example:
Types of discrete probability distributions include:
•Poisson
•Bernoulli
•Binomial
•Geometric

Bernoulli Distribution
The Bernoulli distribution is the discrete probability distribution of a
random variable which takes a binary, Boolean output: 1 with probability
p, and 0 with probability (1-p).

binomial distribution
• The number X of successes in n Bernoulli trials is called a binomial
random variable
• It can be solved by using:
1- Probability function
2- Using table of binomial distribution

Geometric Distribution
The geometric distribution represents the number of failures before you
get a success in a series of Bernoulli trials. This discrete probability
distribution is represented by the probability density function:
f(x) = (1 − p)x − 1p

2nd Scientific Conference of Computer Sciences
Continuous distribution

Some Continuous probability
distributions
■ Many continuous distributions may be used for business applications; two of the
most widely used are:
■ Normal
■ Uniform

The Normal Distribution
• ‘Bell Shaped’
• Symmetrical
• Mean, Median and Mode
are Equal
Location is determined by the mean, μ
Spread is determined by the standard
deviation, σ
The random variable has an infinite
theoretical range:
+  to  
Mean
= Median
= Mode
x
f(x)
μ
σ

The Uniform Distribution
• The uniform distribution is a
probability distribution that has
equal probabilities for all possible
values of the random variable.

The Uniform Distribution

Uniform Distribution
Example: Uniform Probability Distribution
Over the range 2 ≤ x ≤ 6:
2 6
.25
f(x) = = .25 for 2 ≤ x ≤ 6
6 - 2
1
x
f(x)

Data Distribution &
Prepared by:
Maha Sabri Altememe
https://www.mathsisfun.com/data/random-variables-mean-variance.html
Lecture 3
https://docs.google.com/forms/d/1ELHCKg6q_UhjQVcvRh1
eJnElBkvDHus8U9WukyOXfqY/edit?ts=60b1e06d

Definition: Probability Of An Event E.
Suppose that the sample space S = {o1, o2, o3, … oN} has a finite number, N, of
outcomes.
Also each of the outcomes is equally likely (because of symmetry).Then for any event
E  
 
 
  no. of outcomes in
=
total no. of outcomes
n E n E E
P E
n S N
 
Applies only to the special case when
1. The sample space has a finite no.of outcomes,
and
2. Each outcome is equi-probable
If this is not true a more general definition of
probability is required.

Complement of an Event
• Given a set E, the complement of E is the set of
elements that are not in E. The complement is
denoted as E’.
Mutually Exclusive Events
• The sets E1 , E2 ,...,Ek are mutually exclusive if the
intersection of any pair is empty. That is, each
element is in one and only one of the sets E1 , E2 ,...,Ek .

The event occurs if the event A does not occur
A
A
A

Complement
Let A be any event, then the complement of A (denoted by )
defined by:
= {e| e does not belongs to A}
A
A
A
A

Logic:
and are .
A A mutually exclusive
A
A
and S A A
 
   
thus 1 P S P A P A
 
    
 
and 1
P A P A
   
 

Rule The Additive Rule (Mutually Exclusive Events)
i.e.
if A  B = f
(A and B mutually exclusive)
P[A  B] = P[A] + P[B]
P[A or B] = P[A] + P[B]

If two events A and B are are mutually exclusive then:
A B
1. They have no outcomes in common.
They can’t occur at the same time. The outcome of the random
experiment can not belong to both A and B.

Rule The Additive Rule
P[A  B] = P[A] + P[B] – P[A  B]
(In general)
or
P[A or B] = P[A] + P[B] – P[A and B]

P[A  B] = P[A] + P[B] – P[A  B]
A B

B
A
A B

When P[A] is added to P[B] the outcome in A  B are counted twice

       
P A B P A P B P A B
    
Example:
city 1 and city 2 are two of the cities competing for the World university
games. (There are also many others). The organizers are narrowing the
competition to the final 5 cities.
There is a 20% chance that city 1 will be amongst the final 5. There is a
35% chance that city 2 will be amongst the final 5 and an 8% chance
that both city 1 and city 2 will be amongst the final 5. What is the
probability that city 1 or city 2 will be amongst the final 5.

Solution:
Let A = the event that city 1 is amongst the final 5.
Let B = the event that city 2 is amongst the final 5.
Given P[A] = 0.20, P[B] = 0.35, and P[A  B] = 0.08
What is P[A  B]?
Note: “and” ≡ , “or” ≡  .
       
P A B P A P B P A B
    
0.20 0.35 0.08 0.47
   

Conditional Probability
And Independence

A conditional probability is a probability whose sample
space has been limited to only those outcomes that fulfill a
certain condition.
The conditional probability of event A given that event B has
happened is
P(A|B)=P(A ∩ B)/P(B).
The order is very important do not think that
P(A|B)=P(B|A)! THEY ARE DIFFERENT.

Exercise #1
Suppose that A and B are events with probabilities:
P(A)=1/3,
P(B)=1/4,
P(A ∩ B)=1/10
Find each of the following:
1. P(A | B) = P(A ∩ B)/P(B)=1/10/1/4=4/10
2. P(B | A) = P(A ∩ B)/P(A)=1/10/1/3=3/10
3. P(A’| B’) = P(A’ ∩ B’)/P(B’)= P((A U B)’)/(1-P(B))=(1-P(A U
B))/
(1 – P(B))= (1 – (P(A)+P(B)-P(A ∩ B)))/(1-P(B))=
(1 – (1/3+1/4-1/10))/(1-1/10)=(1-29/60)/9/10
= 31/60/9/10=31/54.

Independence Of Events
Two events E and F are said to be independent if and only if
P(E ∩ F)=P(E)P(F).
If the above condition is not satisfied, then we say the two
events E and F are dependent.
When we say two events are independent, we are saying that
if event E has occurred, this will not effect the probability of
event F.
INDEPENDENT EVENTS: The occurrence of one event has
no effect on the probability of the other.

Independent Events
•Two events are independent if the following are true:
P(A|B) = P(A)
P(B|A) = P(B)
P(A AND B) = P(A) ⋅ P(B)
To show 2 events are independent, you must prove one of the above
conditions.

Summary
Probability plays an essential role in our daily life by predicting the possibility of an event,
which is the theory that the statistician uses to help him know how well the random sample
under study represents the community from which the sample is taken. Three important
problems based on the rules of probability are:
1) Knowledge of data types and ways of representing them represented by relative frequency
( in Lecture 1).
2) Methods of estimating such as probability distributions ( in Lecture 2).
3) Calculating the probability in terms of other known probabilities through operations such
as union, intersection, and the laws of probability (in Lecture 3) .

Data Distribution &The Probability Distributions

More Related Content

What's hot (20)

Similar to Data Distribution &The Probability Distributions (20)

Recently uploaded (20)

Data Distribution &The Probability Distributions