# Mathematical Expectation

Mathematical expectation, also known as the expected value, is the summation or integration of a possible values from a random variable.  It is also known as the product of the probability of an event occurring, denoted P(x), and the value corresponding with the actual observed occurrence of the event.  The expected value is a useful property of any random variable.  Usually notated as E(X), the expect value can be computed by the summation overall the distinct values that the random variable can take.  The mathematical expectation will be given by the mathematical formula as, E(X)= Σ (x1p1, x2p2, …, xnpn), where x is a random variable with the probability function, f(x), p is the probability of the occurrence, and n is the number of all possible values    In the case

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.

What is the expected number of coin flips for getting tails?

What is the expect number of coin flips for getting two tails in a row?

Properties and Assumptions:

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).

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).

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 nindependent 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)), where a is a constant and f(X) is the function.

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, where a and b are constants.

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.