1. Probability Distribution

2. Binomial Distribution

3. What is binomial distribution?
We call a distribution a binomial distribution if all of the following are true
There are a fixed number of trials, n, which are all independent.
There must be exactly two mutually exclusive outcomes in a trial, such as True or False, yes or no, success or failure.
The probability of success is the same for each trial.

4. Probability mass function
The binomial distribution is used to obtain the probability of observing x successes in n trials, with the probability of success on a single trial denoted by p.
The formula for the binomial probability mass function is
                                                      
where
                  

5. Some features of binomial distribution
P(S) = p.
P(F) = q = 1-p.
n indicates the fixed number of trials.
x indicates the number of successes (any whole number [0,n]).
p indicates the probability of success for any one trial.
q indicates the probability of failure (not success) for any one trial.
P(x) indicate the probability of getting exactly x successes in n trials.

6. Some statistics
Average = Mean = np
Standard Deviation =

7. Bernoulli distribution
The Bernoulli distribution is a special case of the binomial distribution, where n=1.

8. Example
Suppose that each time you take a free throw shot, you have a 25% chance of making it.  If you take 15 shots,
a. what is the probability of making exactly 5 of them.
Solution
We have
        n  =  15,        r  =  5        p  = .25        q  =  .75
P(5) =  0.165 
There is a 16.5 percent chance of making exactly 5 shots.

9. b.  What is the probability of making fewer than 3 shots?
Solution
The possible outcomes that will make this happen are 2 shots, 1 shot, and 0 shots.  Since these are mutually exclusive, we can add these probabilities.
        P(2)+P(1)+P(0)
        =    =  0.236
There is a 24 percent chance of sinking fewer than 3 shots.

10. Shape of binomial distribution for different values of n and r.
When p is small (<0.5) , the binomial distribution is skewed to the right.
As p increases and approaches to 0.5, the skewness is less noticeable.
When p = 0.5, the binomial distribution is symmetrical.
When p is larger than 0.5, the distribution is skewed to the left.
When n increases binomial distribution approaches to symmetrical.

11. Problems with binomial distribution
The requirement that the probability of the outcome must be fixed overtime is very difficult to meet in practice. In many industrial processes, however, it is extremely difficult to guarantee that this is indeed the case.
Again, the outcome of one trial may affect in any way the outcome of any other trial. Consider an interviewing process in which high-potential candidates are being screened for top positions. If the interviewer has talked to five unacceptable candidates in a row, he may not view the sixth with complete partiality. Therefore, independence is violated.

12. Hypergeometric Distribution

13. Hypergeometric Distribution
An outcome on each trial of an experiment is classified into one of two mutually exclusive categories-a success or a failure
The random variable is the number of successes in a fixed number of trials
The trials are not independent.
We assume that we sample from a finite population without replacement. So, the probability of a success change for each trial.

14. P.M.F of the Hypergeometric Distribution
If X is the number of S’s in a completely random sample of size n drawn from a population consisting of M S’s and (N–M) F’s, then the probability distribution is hypergeometric distribution and is given by :

15. Mean and Variance
The mean and variance of a hypergeometric rv X having pmf h(x;n,M,N) are:

16. Example
In a lot of 20 units out of the production line, 2 units are known to be defective. If the inspector picks a sample of 3 units at random, what is the distribution of the number of defectives in the sample?
What is the probability of having 0, 1, 2 and 3 defectives in the sample?
What is the probability that none of the chosen sample are defective?

17. Poisson Distribution
The Poisson distribution is often used to model the number of occurrences during a given time interval or within a specified region.
The time interval involved can have a variety of lengths, e.g., a second, minute, hour, day, year, and multiples thereof.

18. Some random variables that typically obey the Poisson probability law
The number of misprints on a page (or group of pages) of a book
The number of people in a community living to 100 years of age
The number of wrong telephone numbers that are dialed in a day
The number of customers entering a post office (bank, store) in a give time period
The number of vacancies occurring during a year in the supreme court

19. µ = s2 = l Poisson Distribution
If X is defined to be the number of occurrences of an event in a given continuous interval and is associated with a Poisson process with parameter >0, then X has a Poisson distribution with pdf:
The mean and variance of a Poisson random variable X are:
µ = s2 = l

20. Poisson Distribution
For example, the number of vehicles crossing a bridge in a rural area might be modeled as a Poisson process. If the average number of vehicles per hour, during the hours of 10:00 AM to 3:00 PM is 20 we might be interested in the probability that fewer than three vehicles cross on from 12:30 to 12:45 PM. In this case = (20 per hour)(0.25hours) = 5.