A variable value is determined by the outcome of a random experiment is called a random variable. A random variable is known as chance variable or Stochastic variable. A random variable may be discrete or continuous. If the random variable takes on the integer values such as 0, 1, 2 …. Then it is called a discrete random variable. The number of printing mistakes in each page of a book, the number of telephone calls received by the telephone operator of a firm are examples of discrete random variable. If the random takes on all values, within a certain interval, then the random variable is called a continuous random variable. The amount of rainfall on a rainy day or in a rainy season, the height and weight of individuals are examples of continuous random variable.

In terms of symbols if a variable X can assume discrete set of values X₁. X₂ …. X_k with respective probabilities p₁, p₂ ….. p_k where p₁ + p₂ + … p_k = 1, we say that a discrete probability distribution of X has been defined. The function P(X) which has the respective values p₁, p₂ … for X = X₁, X₂ ….. X_k is called the probability function or frequency function of X.

The probability distribution of a pair of fair dice tossed is given below:

X	2	3	4	5	6	7	8	9	10	11	12
P(X)	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36

Where X denotes the sum of the points obtained. For example, the probability of getting sum 4 is 3/36. Thus in 1200 toss of the dice we would expect 100 tosses to give the sum 4.

It should be noted that a probability distribution is analogous to relative frequency distribution with probabilities replacing relative frequencies. Thus we can think of probability distributions as theoretical or ideal limiting forms of relative frequency distribution when the number of observations is made very large. For this reason, we can think of probability distributions as being distributions for populations, whereas relative frequency distributions are distributions drawn from this population.

Illustration: a dealer in refrigerators estimates from his past experience the probabilities of his selling refrigerators in a day. These are as follows:

No. of refrigerators sold in a day	0	1	2	3	4	5	6
Probability	0.03	0.20	0.23	0.25	0.12	0.10	0.07

Solution: mean number of refrigerators sold

= 0 × 0.03 + 1 × 0.2 + 2 × 0.23 + 3 × 0.25 + 4 × 0.12 + 5 × 0.10 + 6 × 0.07

= 0 + .2 + .46 + .75 + .48 + .5 + .42

= 2.81

Hence the mean number of refrigerators sold in a day is 3.

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