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 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 = 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

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