Random Variable
A variable value is determined by the outcome of a random experiment and is termed as random variable. A random variable is also termed as chance variable or stochastic variable. A random variable can be discrete or continuous. When a random variable takes on the integer values like 0, 1, 2 etc. Then it is termed as discrete random variable. The no. of mistakes in printing in each page of a book, the number of calls received by the telephone operator of a firm is few examples of the discrete random variable. If the random variable takes on all values, within a definite interval, then the random variable is termed as continuous random variable. The quantity of rainfall on a rainy season or in a rainy day, the height & weight of individuals are some examples of continuous random variable.
In terms of symbols, if a variable X can suppose discrete set of values X1, X2 .... Xk with respective probabilities p1, p2 ..... pk where p1 + p2 + ... pk = 1, we say that the discrete probability distribution of X has been defined. The function P(X) which has the particular values p1, p2 ... for X = X1, X2 ..... Xk is termed as the probability function or frequency function of X.
The probability distribution of tossing a pair of fair dice 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 represents the sum of the points obtained. For e.g. the probability of achieving a sum of 4 is 3/36. Therefore in 1200 toss of the dice we would expect 100 tosses to achieve the sum 4.
It must be noted that a probability distribution is analogous to relative frequency distribution with probabilities replacing the relative frequencies. And hence we can think of probability distributions as theoretical or ideal limiting forms of relative frequency distribution when the number of observations is made very big. For this reason, we can think of probability distributions as being distributions for the populations, while the relative frequency distributions are distributions drawn from this population.
Illustration: A dealer in refrigerators estimates from his previous 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
And hence, the mean number of refrigerators sold in a day is 3.