The document discusses theoretical probability distributions and provides examples of random variables that can be modeled by distributions. It introduces common theoretical distributions like the binomial, Poisson, and normal distributions. Specifically, it describes how the binomial distribution can model the probability of outcomes that can succeed or fail with a fixed probability, like the number of goals allowed by a goalkeeper facing a certain number of shots.

1. THEORETICAL PROBABILITY
DISTRIBUTIONS
In day to day life, we come across many random variables such as ------
1.Number of male children in a family having three children.
2.Number of passengers getting into a bus at the bys stand .
3.I.Q. of children
4.Number of stones thrown successively at a mango on the tree until the
mango in hit
5.Marks scored by a candidate in the P.U.E. examination.
For a quick analysis of distributions of such random variables, we
consider their theoretical equivalents. These equivalent distributions are
originated according to certain theoretical assumptions and restrictions.
Such theoretically designed distributions are called theoretical
distributions.

2. • There are many types (families) of theoretical distributions.
Some of them
• (i) Bernoulli distribution
• (ii) Binomial distribution
• (iii) Poisson distribution
• (iv) Hypergeometric distribution
• (v) Normal distribution.
• The Bernoulli distribution and the Binomial distribution were
discovered by James Bernoulli during the first decade of eighteenth
century. These works were published posthumously in 1713.
• The Poisson distribution was introduced by S.D. Poisson in 1837.
• The Normal distribution was introduced by De Moivre in 1753. This
distribution is also called Gaussian distribution.

3. What Does the Binomial
Distribution Describe?
• The probability of getting all “tails” if you
throw a coin three times
• The probability of getting four “2s” if you
roll six dice
• The probability of getting all male
puppies in a litter of 8
• The probability of getting two defective
batteries in a package of six

4. Uses of the Binomial
Distribution
• Quality assurance
• Genetics
• Experimental design

5. The Statistics……..
• If you face 45 shots and allowed 5
goals, your save percentage is .888
• So P(S)=88%
• And P(F)=12%

6. The Problem…………
• What is the probability of saving(P(S)) 70
out of 90 shots?
Probability of Success=
• P(S)= 78%
Probability of Failure=
• P(F)=22%

7. BINOMIAL DISTRIBUTION
• A Probability distribution which has the
following probability mass function (p.m.f)
is
called Binomial distribution.