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) 

 X 1 = { 1 if outcome +, 0 if outcome - }         

 and           

 X 2 = { 1 if outcome +, 0 if outcome - } 

 Then, X = X 1 + X 2 

 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 X i 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}: 

 

 This is only for = and not eqaullity 

 

 

 Where ( n k ) = (n!) / (k! * (n-k)!)

 Where n! = 1 * 2 * 3 * ... n and 0! = 1 

 

 Mean = μ = E [X ] = nπ 

 

 Variance = σ 2 = 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 

 We can construct the standard Z score with: 

 

 We can use the Standard Normal Distribution to approximate a binomial distribution when n is large (say > 25), this is an example of the Central Limit Theorem. The Central Limit Theorem states if you take the sum of a large number of independent, identically distributed variables you can approximate the outcome under a normal distribution. This is the basis of inference in current applied statistics. 

 Poisson Distribution 

 Named after the French mathematician who derived it; the first application was the description of the number of deaths as a result of horse kicking in the Prussian army. It can be used to model the number of events occurring within a given time interval. The probability density (mass) function is: 

 

 where λ is the mean of the distribution (mean number of events); λ determines the shape of the distribution. Other properties which make Poisson distribution popular: 

 

 The mean and variance are both equal to λ 

 The sum of independent Poisson variables is also (!) Poisson variable with mean equal to sum of the individual means 

 Poisson distribution provides an approximation for the Binomial distribution 

 The standard Z score: 

 

 

 If n is large and pi is small, the the Binomial distribution with parameters n and pi can be approximated by a Poisson distribution with mean parameter n*(pi). From there probability calculations with Poisson reduce to probability calculations for a standard normal distribution. 

 When converting a discrete binomial distribution to a continuous distribution we must add correction for continuous conversions: 

 

 Think about it like this: 

 

 In a binomial distribution probability can only accumulate at discrete times {1, 2, 3,...} but since a normal distribution is continuous, you have to account for weather or not you want to include the point. 

 Useful R functions: 

 

 The general rule with functions in R: 

 

 A single point (Probability Density Function) starts with d 

 Cumulative Distribution starts with p 

 Quantile starts with q 

 Random variables start with r