Some Special Distributions part1

Bernoulli Distribution
A Bernoulli experiment is a random experiment in which the outcomes can be classified in one of two mutually exclusive and exhaustive ways. success/failure.
The success has a fixed probability of success p, and the failure has
probability of failure q=1-p.
That is X be a random variable associated with a Bernoulli trial.
? X = 1 implies a success.
? X = 0 implies a failure.
The then pmf of X can be written as :
p(x)=px (1-p)1-x x=0,1

Binomial Distribution
• Binomial experiment has:
1) Independent and identical Bernoulli n trials.
2) Probability of success p is fixed for each trial.
3) Only two possible outcomes for each trial.
4) Number of trails are fixed.
The random variable X which counts the number of success of a Binomial experiment is said to have a Binomial distribution with parameters n and p and its pmf .
Negative Binomial Distribution
• Consider a sequence of independent repetitions of a random experiment with constant probability p of success. Let the random variable Y denote the total number of failures in this sequence before the rth success that is, Y +r is equal to the number of trials required to produce exactly r successes. The pmf of Y is called a Negative Binomial distribution