This document discusses probability distributions, including binomial and Poisson distributions. It defines key terms like random variables, discrete and continuous probability distributions, and the assumptions and constants of binomial distributions. Specifically, it explains that a binomial distribution describes experiments with two possible outcomes (success and failure) where there is a fixed number of trials, the probability of success is the same for each trial, and trial results are independent. The mean of a binomial distribution is np and the variance is npq, where n is the number of trials, p is the probability of success, and q is the probability of failure.

1. Probability Distributions
Binomial and Poisson

2. Probability Distribution
• Listing of all possible outcomes of an experiment
together with their probabilities
• Ex Toss of fair coin 2 times
OUTCOME PROBABILITY
TT ¼
TH ¼
HT ¼
HH ¼
What is Probability Distribution of number of heads

3. Probability Distribution
Number of Heads P(x)
0 ¼
1 ½
2 ¼
Sum of all probabilities =1

4. Random Variable
• Random variable is a real number ‘x’
associated with the outcome of a random
experiment
• Random Variable ‘x ‘ is of 2 types
- Discrete Random Variable
- Continuous Random Variable

5. Discrete & Continuous Random
Variable
• Discrete Random Variable – ‘x’ can take values
which are countable and can be denoted as a
‘number ‘ .
‘x’ is 0,1,2,3,4………..
ex number of defectives in sample of ‘k’ items
• Continuous Random Variable – ‘x’ can take any
values between 2 given numbers and can have
infinite number of possible values in that range
ex possible weights , heights , temperature ,
distance in a given interval

6. Discrete Probability Distribution
• A discrete random variable assumes each of
its values with a certain “probability”
• A table listing all possible values that discrete
random variable can take along with
associated probabilities is called “ Discrete
Probability Distribution”

7. Types of Discrete Probability
Distribution
Two types
• Binomial Distribution
• Poisson Distribution

8. Continuous Probability Distribution
• Normal Distribution

10. Binomial Distribution
• Experiments with only 2 outcomes are
“Binomial Experiments "and the two
outcomes are called ‘success (p)’ and ‘failure
(q)’
ex toss of a coin, throw of dice – either even
or odd number will come , either vote for a
candidate or don’t vote for that candidate

11. Binomial Distribution
1 . If one outcome appears , the other cannot appear – two outcomes
are ‘mutually exclusive’
2.Since there is no other possibility, these two outcomes are
‘collectively exhaustive’
3. Since two outcomes are ‘mutually exclusive’ and ‘exhaustive’ , the
sum of their individual probabilities is equal to 1
p+ q=1
p= probability of success
q=probability of failure
Probability distribution of success is called ‘Binomial Distribution”

12. Assumptions in Binomial Experiments
• Process in which each trial (experiment) can
result in one of 2 states is called “Bernoulli or
Binomial Process”
1 .Each trial has 2 possible outcomes called
‘success ‘ and ‘failure’
2.There is finite number of trials ‘n’
3.All trials are identical i.e. probability of each
trial is same
4.Trials are independent of each other

13. Constants of Binomial Distribution
• Mean = np
• Variance = npq

16. Problem

17. Solution

18. Solution

19. Solution

20. Solution

21. Problem
• If a new drug is found to be effective 40% of
the time , then what is the probability that in
a random sample of 4 patients , it will be
effective on 2 of them

22. Solution