Statistical Learning - Probability and Distributions
Probability – Meaning &
Concepts
Probability refers
to chance or likelihood of a particular event-taking place.
An event is an outcome of an experiment.
An experiment is a process that is performed to understand and observe
possible
outcomes.
Set of all
outcomes of an experiment is called the sample
space.
Example
• In a manufacturing unit three parts
from the assembly are selected. You are observing whether they are defective or
non-defective. Determine
a)
The sample space.
b)
The event of
getting at least two defective parts.
Definition of Probability
Marginal Probability
• Contingency table consists of rows and columns of two attributes at different levels with frequencies or numbers in each of the cells. It is a matrix of frequencies assigned to rows and columns.
• The term marginal is used to indicate that the probabilities are calculated using a contingency table (also called joint probability table).
Solution
a)
What
is the probability that a randomly selected family is a buyer of the
Car?
•
80/200 =0.40.
b)
What
is the probability that a randomly selected family is both a buyer of car and
belonging to income of Rs. 10 lakhs and above?
•
42/200 =0.21.
c)
A
family selected at random is found to be belonging to income of Rs 10 lakhs and
above. What is the probability that this family is buyer of car?
• 42/80 =0.525. Note this is a case of
conditional probability of buyer given income is Rs. 10 lakhs and above.
Bayes’ Theorem
• Bayes’ Theorem is used to revise
previously calculated probabilities based on new information.
•
Developed by
Thomas Bayes in the 18th Century.
•
It is an
extension of conditional probability.
Many modern machine
learning techniques rely on Bayes'
theorem. For instance, spam filters use Bayesian updating to determine whether
an email is real or spam, given the words in the email. Additionally, many
specific techniques in statistics, such as calculating p-values or interpreting medical
results, are best described in terms of
how they contribute to updating hypotheses using Bayes'
theorem.
What is a Probability
Distribution
• In precise terms, a probability distribution is a total
listing of the various values the random variable can take along with the
corresponding probability of each value. A real life example could be the
pattern of distribution of the machine breakdowns in a manufacturing unit.
• The random variable in this example
would be the various values the machine breakdowns could assume.
• The probability corresponding to each
value of the breakdown is the relative frequency of occurrence of the
breakdown.
• The probability distribution for this
example is constructed by the actual breakdown pattern observed over a period
of time. Statisticians use the term
“observed distribution” of breakdowns.
Binomial Distribution
• The Binomial Distribution is a widely
used probability distribution of a discrete random variable.
• It plays a major role in quality control and quality assurance function.
Manufacturing units do use the binomial distribution for defective analysis.
• Reducing the number of defectives
using the proportion defective control chart (p chart) is an accepted practice
in manufacturing organizations.
• Binomial distribution is also being
used in service organizations like
banks, and insurance corporations to get an idea of the proportion customers
who are satisfied with the service quality.
Conditions
for Applying Binomial Distribution
(Bernoulli Process)
• Trials are independent and random.
•
There are fixed
number of trials (n trials).
• There are only two outcomes of the
trial designated as success or failure.
•
The probability
of success is uniform through out the n trials
Example for
Binomial Distribution
A bank issues credit cards to customers
under the scheme of Master Card. Based on the past data, the bank has found out
that 60% of all accounts pay on time following the bill. If a sample of 7
accounts is selected at random from the current database, construct the
Binomial Probability Distribution of accounts paying on time.
No comments:
Post a Comment