. Mathematical Expectation One of the important characteristics of a random variable is its expectation. Synonyms for expectation are expected value, mean, and first moment. The definition of mathematical expectation is driven by conventional idea of numerical average.
The numerical average of n numbers, say a a a a n is The average is used to summarize or characterize the entire collection of n numbers , with single value. Illustration . Consider ten numbers , , , , , , , , , The average is If ten numbers , , , , , , , , , − are considered as the values of a random variable X the probability mass function is given by – The above calculation for average can also be rewritten as . - - This illustration suggests that the mean or expected value of any random variable may be obtained by the sum of the product of each value of the random variable by its corresponding probability.
So average (value of x ) × (probability) This is true if the random variable is discrete. In the case of continuous random variable, the mathematical expectation is essentially the same with summations being replaced by integrals. Two quantities are often used to summarize a probability distribution of a random variable X . In terms of statistics one is central tendency and the other is dispersion or variability of the probability distribution.
The mean is a measure of the centre tendency of the probability distribution, and the variance is a measure of the dispersion, or variability in the distribution. But these two measures do not uniquely identify a probability distribution. That is, two different distributions can have the same mean and variance. Still, these measures are simple, and useful in the study of the probability distribution of X .