Discrete vs Continuous Probability Distributions
Statistical experiments are random experiments that can be repeated indefinitely with a known set of outcomes. A variable is said to be a random variable if it is an outcome of a statistical experiment. For example, consider a random experiment of flipping a coin twice; the possible outcomes are HH, HT, TH, and TT. Let the variable X be the number of heads in the experiment. Then, X can take the values 0, 1 or 2, and it is a random variable. Observe that there is a definite probability for each of the outcomes X = 0, X = 1, and X = 2.
Thus, 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 f is called the probability mass/density function of the random 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, 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 cumulative distribution 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; F(a) = 1, if a≥2.
What is a discrete probability distribution?
If the random variable associated with the probability distribution is discrete, then such a probability 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 random variable X can have only a finite number of values. Common examples of discrete probability 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 probability distribution?
If the random variable associated with the probability distribution is continuous, then such a probability distribution is said to be continuous. Such a distribution is defined using a cumulative distribution function (F). Then it is observed that the probability density function ƒ(x) = dF(x)/dx and that ∫ƒ(x) dx = 1. Normal distribution, student t distribution, chi squared distribution, and F distribution are common examples for continuous probability distributions.
What is the difference between a discrete probability distribution and a continuous probability distribution?
• In discrete probability distributions, the random variable associated with it is discrete, whereas in continuous probability distributions, the random variable is continuous.
• Continuous probability distributions are usually introduced using probability density functions, but discrete probability distributions are introduced using probability mass functions.
• The frequency plot of a discrete probability distribution is not continuous, but it is continuous when the distribution is continuous.
• The probability that a continuous random variable will assume a particular value is zero, but it is not the case in discrete random variables.