# Probability Distributions

## Probability Distributions

There are two general types of probability distribuitons: Discrete and Continuous

Discrete Probability Distributions are those that are defined across a countable number of possible values. A probability mass function is used to describe how the total probability (sum = 1) is distributed across the set of possible values. The cumulative distribution function is discontinuous, jumping at each of the possible values.

Continuous Probability Distributions are those that have continuous cumulative distribution functions. Each outcome is possible, but technically, its probability is zero. The derivative of the cumulative distribution function is the probability density function.

It is possible to concoct a distribution that has features of both discrete and continuous probability distributions, but we'll not concern ourselves with those just yet. Here are lists of the most "popular" discrete and continuous distributions:

## Distribution Fitting

Using R, the package "MASS" includes a function "fitdistr," which finds maximum likelihood univariate distributions.  To use the package,

1. Request the package with the R command install.packages("MASS")
2. Select a nearby CRAN site from which to download the package.
3. Attach the package with the R command attach("MASS")
4. Get help with the fitdistr function by entering the R command \$fitdistr

An example:

Z <- rnorm(100,2,3)

Fit <- fitdistr(Z,"normal")

Fit       # Returns estimates of mean & sd, together with their standard errors.  Estimates should be near true values mean = 2 and sd = 3, respectively.

summary(Fit)    # Reveals that other info are available.  n = sample size and loglik = log likelihood

Fit\$loglik          # Returns the log likelihood

NIST/SEMATEC Handbook of Engineering Statistics:

Wolfram MathWorld. The following link distributions of the two types:

• Discrete Distribution:
• Continuous Distribution: