Discrete vs Continuous Distributions
The distribution of a variable is a description of the frequency of occurrence of each possible outcome. A function can be defined from the set of possible outcomes to the set of real numbers in such a way that ƒ(x) = P(X = x) (the probability of X being equal to x) for each possible outcome x. This particular function ƒ is called the probability mass/density function of the variable X. Now the probability mass function of X, in this particular example, can be written as ƒ(0) = 0.25, ƒ(1) = 0.5, and ƒ(2) = 0.25.
Also, a function called cumulative distribution function (F) can be defined from the set of real numbers to the set of real numbers as F(x) = P(X ≤ x) (the probability of X being less than or equal to x) for each possible outcome x. Now the probability density function of X, in this particular example, can be written as F(a) = 0, if a<0; F(a) = 0.25, if 0≤a<1; F(a) = 0.75, if 1≤a<2 and F(a) = 1, if a≥2.
What is a discrete distribution?
If the variable associated with the distribution is discrete, then such a distribution is called discrete. Such a distribution is specified by a probability mass function (ƒ). The example given above is an example of such a distribution since the variable X can have only a finite number of values. Common examples of discrete distributions are binomial distribution, Poisson distribution, Hyper-geometric distribution and multinomial distribution. As seen from the example, cumulative distribution function (F) is a step function and ∑ ƒ(x) = 1.
What is a continuous distribution?
If the variable associated with the distribution is continuous, then such a distribution is said to be continuous. Such a distribution is defined using a cumulative distribution function (F). Then it is observed that the density function ƒ(x) = dF(x)/dx and that ∫ƒ(x) dx = 1. Normal distribution, student t distribution, chi squared distribution, F distribution are common examples for continuous distributions.
What is the difference between discrete distribution and continuous distribution?
• In discrete distributions, the variable associated with it is discrete, whereas in continuous distributions, the variable is continuous.
• Continuous distributions are introduced using density functions, but discrete distributions are introduced using mass functions.
• The frequency plot of a discrete distribution is not continuous, but it is continuous when the distribution is continuous.
• The probability that a continuous variable will assume a particular value is zero, but it is not the case in discrete variables.