# Statistics/Distributions/Discrete

'Discrete' data are data that assume certain discrete and quantized values. For example, true-false answers are discrete, because there are only two possible choices. Valve settings such as 'high/medium/low' can be considered as discrete values. As a general rule, if data can be counted in a practical manner, then they can be considered to be discrete.

To demonstrate this, let us consider the population of the world. It is a discrete number because the number of civilians is theoretically countable. But since this is not practicable, statisticians often treat this data as continuous. That is, we think of population as within a range of numbers rather than a single point.

For the curious, the world population is 6,533,596,139 as of August 9, 2006. Please note that statisticians did not arrive at this figure by counting individual residents. They used much smaller samples of the population to estimate the whole. Going back to Chapter 1, this is a great reason to learn statistics - we need only a smaller sample of data to make intelligent descriptions of the entire population!

Discrete distributions result from plotting the frequency distribution of data which is discrete in nature.

### General Properties

#### Cumulative Distribution Function

A discrete random variable has a cumulative distribution function that describes the probability that the random variable is below the point. The cumulative distribution must increase towards 1. Depending on the random variable, it may reach one at a finite number, or it may not. The cdf is represented by a capital F.

ń====Probability Mass Function====

A discrete random variable has a probability mass function that describes how likely the random variable is to be at a certain point. The probability mass function must have a total of 1, and sums to the cdf. The pmf is represented by the lowercase f.

#### Special Values

The expected value of a discrete variable is ${\displaystyle \sum _{n_{min}}^{n_{max}}x_{i}f(x_{i})}$

The expected value of any function of a discrete variable g(${\displaystyle X}$) is ${\displaystyle \sum _{n_{min}}^{n_{max}}g(x_{i})f(x_{i})}$

The variance is equal to ${\displaystyle E((X-E(X))^{2})}$