# Probability Distribution of a Random Variable  ## Probability Distribution of a Random Variable:

In statistics, we had sufficient discussions about frequency distribution, which were based on observations. In a frequency distribution, the frequencies for different values of the variable under consideration are based on actual observation. Thus, if an unbiased coin is tossed 20 times, we may get head 13 times, though theoretically, we shall expect head 10 times. But 13 is the observed frequency here.

In this section, we shall discuss probability distribution based on theoretical considerations. A symbol that can assume any of the prescribed set of values is known as a variable. We consider S to be the Sample Space of some given random experiment. The outcomes, i.e., the points of the sample space are not always numbers. But we may assign a real number to each sample point according to some definite rule, which gives us a function defined on the sample space S. This real-valued function defined on the sample space of an experiment is known as a random variable. The values of a random variable are real numbers connected with the outcomes of an experiment.

In the random experiment of tossing 3 coins, the sample space is given by-

S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT }

If x be the random variable denoting the ‘number of heads’ in this case, then we assign a number to each sample point as follows:

One can define many other random variables on the same sample space. For example, if x denotes the random variable defined as the ‘square of the number of tails’ in the above experiment, then we have

In the first example, the random variable x can assume only four discrete values 0, 1, 2, and 3 and in the second example, x can assume the values 0, 1, 4, and 9. The random variables defined in these two cases are called discrete random variables. We can define a discrete random variable as the one which can assume only finite number of values. If, however, the random variable assumes any value between certain limits, then it is known as a continuous random variable.

The required probability distribution is-

The required probability distribution is-

The required probability distribution is-

The required probability distribution is-

The required probability distribution is-