Module 4: Discrete Distributions
For any domain there are infinitely many distributions. The most common and famous distributions get a name; Binomial, Negative Binomial, Geometric, Hypergeometric, Poisson, etc. In this section we focus on Binomial and Poisson distributions.
The Bernoulli Distribution is a special member of the distribution family. It is the simplest example of a Binomial distribution, with only two domains (aka dicothomous distribution). A experiment which only has two domains is called a Bernoulli experiment. Ex. the number of students who get an A on a test, whether a person has a disease or not.
If we have two Bernoulli independent trials with equal probability of a positive result, we refer to that probability as pi (not 3.14)
X1 = { 1 if outcome +, 0 if outcome - } and X2 = { 1 if outcome +, 0 if outcome - }
Then, X = X1 + X2
The variable X above is a random variable with domain of {0, 1, 2} as it is a result of the two trials. The distribution is an example of a Binomial (2, pi) distribution.
More generally, if Xi are n Bernoulli independent trials with probability of a positive result equals pi
The domain of X a Binomial (n, pi) is {0, 1, 2... n}. When n=1 the binomial reduces to Bernoulli
For k in domain {0, 1, 2, ...n}:
- P{X = k} = (nk) = (n!) / (k! * (n-k)!) Where n! = 1 * 2 * 3 * ... n and 0! = 1
- Mean = μ = E [X ] = nπ
- Variance = σ2 = Var [X] = nπ(1 − π)
Note that variance is a function of mean, Mean > Variance and for a fixed n the variance is maximum at pi = .5