In this article, we are going to look into random variables in statistics, grab your pen and paper. You’re going to need it.

Consider an experiment of tossing two coins with the corresponding sample space S = { HH, HT, TH, TT }.

Suppose interest is on the total number of heads that will be obtained. Let X represent the total number of heads that will be obtained.

Using the sample space S defined above, X will assume the values 0,1,2. A numerical valued variable such as this is an example of what is usually called a random variable.

Discrete Random Variables

Having defined random variables X, the next interest is usually that of calculating the probability that x will assume certain specified value within the range of all possible values.

For example, In the experiment of rolling two dice earlier described, If X represents the sum of the numbers obtained, an experimenter may be interested in calculating the probability that the sum is 5, the sum is less than 7.

A sample space that consists of a finite number or a countably infinite sequence of points is called a discrete sample space, whereas one that contains one or more intervals of points is called a continuous sample space.

A random variable defined on a discrete sample space that assumes only a  finite number or a countably infinite sequence of values is called a discrete random variable.

Continuous Random Variables

Continuous random variables are usually associated with experiments that involve some type of measurements. with weights, lengths, heights, temperatures, velocities, etc are usually considered continuous.

As should be expected, a continuous random variable is one that can assume any value in some intervals of values and for which the probability is zero that is will not assume any particular value in the interval.

Density Functions

In order to calculate probabilities associated with discrete or continuous random variables, we shall introduce a function called discrete probability density function (for discrete random variables) and probability density function (for continuous random variables)

Joint Density Functions

There are several experiments in which several random variables are involved rather than just one.

To illustrate this situation, consider two random variables X and Y.

The function (x, y) which gives the probability that X will assume the value X and at the same time Y will assume the value y is called the joint density function for the random variables, X and Y.


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