Mathematical expectation is the expected value of random phenomenon, also sometimes called an expected value. After the probability distribution of a random variable has been constructed, the next task is to calculate the mean of that random variable. In other words, one can say that the mathematical expectation of the random variable is computed. In the case of a discreet random variable, the mathematical expectation will be given by the mathematical formula as, E(X)=and in the case of continuous random variable, the mathematical expectation will be given by the mathematical formula as, E(X)=. In the discreet case of mathematical expectation formula, p(x) denotes the probability mass function and f(x), in the continuous case of mathematical expectation formula, denotes the probability density function.
The mathematical expectation of an indicator variable can be zero if there is no occurrence of an event A, and the mathematical expectation of an indicator variable can be one if there is an occurrence of an event A. Thus, it is a useful tool to find the probability of event A.
Properties:
The first property is that if X and Y are the two random variables, then the mathematical expectation of the sum of the two variables is equal to the sum of the mathematical expectation of X and the mathematical expectation of Y, provided that the mathematical expectation exists. In other words, E(X+Y)=E(X)+E(Y).
This property is valid in the case of ‘n’ number of variables. The generalization of this property states that the mathematical expectation of the sum of ‘n’ random variables is equal to the sum of the mathematical expectation of all the ‘n’ random variables provided that their mathematical expectation exists.
The second property is that the mathematical expectation of the product of the two random variables will be the product of the mathematical expectation of those two variables, provided that the two variables are independent in nature. In other words, E(XY)=E(X)E(Y).
This property is valid in the case of ‘n’ number of variables. The generalization of this property states that the mathematical expectation of the product of the ‘n’ number of independent random variables is equal to the product of the mathematical expectation of the ‘n’ independent random variables.
The third property states that the mathematical expectation of the product of a constant and the function of a random variable is equal to the product of the constant and the mathematical expectation of the function of that random variable provided that their mathematical expectation exists. The third also states that the mathematical expectation of the sum of a constant and the function of a random variable is equal to the sum of the constant and the mathematical expectation of the function of that random variable provided that their mathematical expectation exists. In other words, E(a f(X))=a E(f(X)) and E(a+f(X))=a+E(f(X)).
The fourth property states that the mathematical expectation of the sum of the product between a constant and the function of a random variable and the other constant is equal to the sum of the product between the constant and the mathematical expectation of the function of that random variable and the other constant provided that their mathematical expectation exists. In other words, E(aX+b)=aE(X)+b.
The fifth property states that the mathematical expectation of the linear combination of the random variables is equal to the sum of the product between the ‘n’ constant and the mathematical expectation of the ‘n’ number of variables. In other words, E(∑aiXi)=∑ ai E(Xi). Here, ai, (i=1…n) are constants.
