M3_Statistics foundations for business analysts_Presentation.pdf

Probability
Probability is the numerical measure of
the likelihood that an event will occur.

Probability - Concepts
Sample Space
Sample space of an experiment or random trial is
the set of all possible outcomes or results of that
experiment.

Event
An event is a set of outcomes of an experiment (a
subset of the sample space) to which a probability
is assigned
For dice face showing ‘5’ –>
E = { 5 }
For dice face showing greater than 3 value ->
E = { 4, 5, 6 }

Complement of
an Event
Given an event A, the complement of A is defined
to be the event consisting of all outcomes that are
not in A.
P(A) = 1 - P(Ac )
Probability of getting a project –> 0.4
Probability of not getting a project -> 1-0.4 =0.6
P(Ac)
P(A)

Addition Law
The addition law provides a way to compute the
probability that event A or event B occurs or both
events occur.
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)
A or B A B Both A and B
Probability of getting project A –> 0.4
Probability of getting project B –> 0.7
Probability of getting both project -> 0.2
Probability of getting any project -> 0.4 + 0.7 - 0.2= 0.9

Probability
Distribution
A probability distribution describes the uncertainty
of a numerical outcome.
Project time Probability
3 10.0%
4 25.0%
5 30.0%
6 20.0%
7 15.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
3 4 5 6 7
probability

Probability
Distribution
A probability distribution describes the uncertainty
of a numerical outcome.
Sample Space Probability
Head 0.50
Tail 0.50
0.00
0.10
0.20
0.30
0.40
0.50
0.60
Head Tail
Sample Space Probability
1 16.67%
2 16.67%
3 16.67%
4 16.67%
5 16.67%
6 16.67% 0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
18.00%
1 2 3 4 5 6

Summary
Measures of a
Probability
Distribution
1. Mean of a Probability Distribution, 𝜇
2. Variance of a Probability Distribution, 𝜎2
3. Standard Deviation of a Probability Distribution, 𝜎

Discrete vs
Continuous
Random variable is a numerical description of the
outcome of a random experiment
10% 25% 30% 20% 15%
0%
5%
10%
15%
20%
25%
30%
35%
3 4 5 6 7
Completion Time (Months)
Probablity
Discrete random variable
Probability mass distribution
Continuous random variable
Probability density distribution

Use of
probability
distribution
functions
10% 25% 30% 20% 15%
0%
5%
10%
15%
20%
25%
30%
35%
3 4 5 6 7
Completion Time (Months)
Probablity

Probability Distribution
Discrete Uniform
The possible values of the probability mass function,
f(x), are all equal
F(x) = 1/n
n = the number of unique values of random variable
Eg: Rolling a single fair die
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
1 2 3 4 5 6

Discrete
Binomial
The binomial distribution is a discrete distribution that
can occur in two situations:
1. When performing a sequence of identical
experiments, each of which has only two possible
outcomes. Eg. Heads in tossing coin 5 times
2. When sampling from a population with only two
types of members. Eg. Male students in a class of
100 students

Discrete
Binomial
How many customers will show up at restaurant if we
are sending a mail to n (say 100) people. Find
probability of exactly 20 customers.
Important Parameters:
1. Probability of success on each trial – p
2. Number of independent, identical trials – n
3. Number of success - x
1. where

Discrete Poisson
It usually applies to the number of events occurring within a
specified period of time or space. Examples
1. A store manager is studying the arrival pattern to the store.
The events are customer arrivals, the number of arrivals in
an hour is Poisson distributed.
2. A retailer is interested in the number of customers who
order a particular product in a week.
3. A civil engineer is interested in the numbers of potholes in a
10 km stretch of road
The Poisson distribution is characterized by a single parameter,
usually labelled 𝜆 (Lambda) or 𝜇 . It is both the mean and the
variance of the Poisson distribution

Discrete Poisson
The Poisson distribution is characterized by a single parameter,
usually labelled 𝜆 (Lambda) or 𝜇 . It is both the mean and the
variance of the Poisson distribution

Discrete Poisson
John –Manager at Sukuzi Motors
Johns wants your help in deciding the car inventory level
Historical average demand per month is 13 car
You can help John by providing him with the probability
distribution of the demand of cars

Continuous
Distribution
• Continuous random variable
• Probability density function
• Probabilities of interval rather than a point
• Probability = Area under the curve

Uniform
Probability
Distribution

Uniform
Probability
Distribution
A chocolate bar can weight
anywhere between 120 gm to 140
gm. Bars weighing >140 gm or <
120 gm are rejected
Probability of less than 125 gm
=
125 − 120
140 − 120
=
5
20
= 0.25
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 =
120 + 140
2
= 130
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 =
(140 − 120)2
12
=
400
12
= 33.33

Normal
Distribution
• Single most important distribution in statistics
• Used for wide variety of practical applications in which the random
variables are heights and weights of people, test scores, scientific
measurements, amounts of rainfall, and other similar values
• Also widely used in business applications to describe uncertain quantities
such as demand for products, the rate of return for stocks and bonds, and
the time it takes to manufacture a part or complete many types of service-
oriented activities such as medical surgeries and consulting engagements

Normal
Distribution
Key
Characteristics
1. Two important parameters: the mean 𝜇 and the standard deviation 𝜎.
2. Highest point on the normal curve is at the mean (negative, zero, or positive)
3. The normal distribution is symmetric
4. The tails of the normal curve extend to infinity in both directions

Normal
Distribution
Key
Characteristics
5. Probabilities for the normal random variable are given by areas under the
normal curve.
6. The standard deviation determines how flat and wide the normal curve is.

Normal
Distribution
Key
Characteristics
7. The percentages of values in some commonly used intervals are
• 68.3% of the values are within plus or minus one 𝜎 of its 𝜇.
• 95.4% of the values are within plus or minus two 𝜎 of its 𝜇.
• 99.7% of the values are within plus or minus three 𝜎 of its 𝜇.

Normal
Distribution
Key
Characteristics

Normal
Distribution
Example
Edison motor is an electric automobile company, they have launched their state-
of-art long range electric truck eT-90. To promote sales, they are providing 10
years replacement warranty on the battery system. For testing and simulation,
they assumed that the average life of motor system 4500 days with standard
deviation of 600 days. What percentage of customers will get replacement?
600
4500
3650

Normal
Distribution
Example
Edison motor is an electric automobile company, they have launched their state-
of-art long range electric truck eT-90.. For testing and simulation, they assumed
that the average life of motor system 4500 days with standard deviation of 600
days. What should be the warranty period to cover only 15 % of the customers
600
4500
15 %

Exponential
Distribution
• Discrete Poisson distribution usually applies to the number of events
occurring within a specified period of time or space
• An alternative way to view the uncertainty in this process is to consider the
times between consecutive events.
• The most common probability distribution used to model these times,
often called interarrival times, is the exponential distribution.

Exponential
Distribution
• Probability density function
OR

Exponential
Distribution
Suppose John is a manager for an airline company. Customers have to collect
their boarding pass from the counter before boarding . The processing time
for generating a boarding pass is 3 minutes. Expected number of Business
class arrivals per hour is 6. What is the probability that the next business class
customers has to wait to collect his/her boarding pass

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