“Theoretical” Reference Distributions

There are a number of theoretical (i.e. based on some kind of theory or understanding), as opposed to empirical (i.e. based on direct experience) distributions (or descriptions of the frequency at which different values of a variable are likely to occur) that can be used as reference distributions for specific statistics obtained from particular data sets.  These distributions can be described using their probability density functions (or pdfs), which are analogous to the histogram, and cumulative density functions (or cdfs), which are analogous to the cumulative frequency curve.

Uniform distribution

The uniform distribution is defined for a particular interval on the number line, say  (read as “the interval from a to b”), such that the probability of observing and value within this interval is p, while the probability of observing outside of this interval is zero.  Let  represent a random variable (and let  be the i-th observation of athe random variable , or a variable whose observed values are determined by chance.  If  is continuous (i.e., it can take on any value, not just integer values), then the pdf of the uniform distribution is

Binomial distribution

The binomial distribution is a discrete distribution that applies to situations where only two outcomes are possible (e.g. rain, no rain).  The pdf of the binomial distribution is

Poisson distribution

The Poisson distribution is a discrete distribution that illustrates the probability of observing a particular number of events in an area or time interval, when the mean number of events per area or time are known.  The pdf of the Poisson distribution is

Normal distribution

The normal distribution arises frequency in practice as a consequence of the Central Limit Theorem, and the fact that many phenomena that are observed in practice represent integration of processes over time or space.  The normal distribution is a continuous distribution, and its pdf is given by

The standard normal distribution applies to the simple case when  and   The pdf of the standard normal distribution is