Discrete vs. Continuous Probability Distributions

All probability distributions can be classified as discrete probability distributions or as continuous probability distributions, depending on whether they define probabilities associated with discrete variables or continuous variables.

Discrete vs. Continuous Variables :

If a variable can take on any value between two specified values, it is called a continuous variable ; otherwise, it is called a discrete variable.

Discrete Probability Distributions

If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution.

An example will make this clear. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the random variable X represent the number of Heads that result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable.

The probability distribution for this statistical experiment appears below.

Number of heads Probability

   0                          0.25

   1                          0.50

   2                          0.25

The above table represents a discrete probability distribution because it relates each value of a discrete random variable with its probability of occurrence. 

Discrete probability distributions types.

Binomial probability distribution

Hypergeometric probability distribution

Multinomial probability distribution

Negative binomial distribution

Poisson probability distribution

Note: With a discrete probability distribution, each possible value of the discrete random variable can be associated with a non-zero probability. Thus, a discrete probability distribution can always be presented in tabular form.

If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution.

Continuous Probability Distributions :

A continuous probability distribution differs from a discrete probability distribution in several ways.

The probability that a continuous random variable will assume a particular value is zero.

As a result, a continuous probability distribution cannot be expressed in tabular form.

Instead, an equation or formula is used to describe a continuous probability distribution.

Most often, the equation used to describe a continuous probability distribution is called a probability density function. Sometimes, it is referred to as a density function, a PDF, or a pdf. For a continuous probability distribution, the density function has the following properties:

Since the continuous random variable is defined over a continuous range of values (called the domain of the variable), the graph of the density function will also be continuous over that range.

The area bounded by the curve of the density function and the x-axis is equal to 1, when computed over the domain of the variable.

The probability that a random variable assumes a value between a and b is equal to the area under the density function bounded by a and b.

For example, consider the probability density function shown in the graph below. Suppose we wanted to know the probability that the random variable X was less than or equal to a. The probability that X is less than or equal to a is equal to the area under the curve bounded by a and minus infinity - as indicated by the shaded area.

Note: The shaded area in the graph represents the probability that the random variable X is less than or equal to a. This is a cumulative probability.

Continuous probability distributions types :

Normal probability distribution

Student's t distribution

Chi-square distribution

F distribution