Statistical Methods

Unit 2 - Probability distributions

2.3.Probability distributions
There are two types of probability distributions depending upon the type variables under study. It may be discrete and continuous. Probability distribution is said to be discrete if it is based on a discrete random variable and continuous when it is based on a continuous random variable. A mass probability distribution for discrete random variable is a listing of all possible values with respective probabilities of occurrence.

Characteristics of probability distribution:

• Probabilities are non-negative i.e. P(x) ≥O
• Sum of probabilities of all values of random variable is equal to unity. i.e., ∑P(x) = 1.

As discrete random variable can take only a finite number of values or a countable infinite number of values, it is possible to list all the values with the corresponding probabilities. The probability distribution of a discrete random variable is called probability mass function.

In the case of continuous random variable, it is no longer meaningful to list all the values with the corresponding probabilities and hence the probability of a random variable falling in a given interval is listed. A histogram can be drawn taking probability on Y axis with large number of small intervals of random variable on X axis. A smooth curve passing through upper sides of the rectangles of the histogram can be drawn. In many cases it is possible to determine a function f(x) that approximates the curve. This function is called a probability density function. Here also the two basic conditions should be satisfied (i) f(x) ≥ 0 and (ii) the total area under the curve f(x) and the x-axis is equal to 1.

Example1
Find the probability distribution of an outcome in throwing of a dice experiment.
Answer
Let X denote the outcome of the experiment. Then the probability distribution of X is given by,

Note that ∑P(X) = 1