Mathematical Expectation
The mathematical expectation of a particular random phenomenon basically means the average value of the random phenomenon.
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In statistics, there are basically two types of random variable, namely discrete and continuous random variables. According to the type of the random variable, the formula for the mathematical expectation is given. So, for the discrete type of random variable, it is defined as the sum of the product of the random variables and the probability mass function (pmf) of those random variables. The formula for this is the following:
E(X) = for discrete random variable.
Similarly, for the continuous type of random variable, the mathematical expectation is given by the integral of the product of the random variables and the probability density function (pdf) of those random variables. The formula for this is the following:
E(X) = , for the continuous random variable.
There are certain properties of mathematical expectation:
The first property is that of the additional theorem. This property states that if there is an X and Y, then the sum of those two random variables are equal to the sum of the mathematical expectation of the individual random variables. In other words,
E(X+Y) = E(X) + E(Y), provided that all the expectations exist.
This property of the mathematical expectation also has the generalized form, which states that the sum of the ‘n’ number of random variables is equal to the sum of the mathematical expectation of the individual ‘n’ number of random variables. In other words,
E(X1+X2+ …… + Xn) = E(X1) +E(X2)+ …… +E(Xn), provided that all the expectations exist.
The second property is that of the multiplication theorem.
This property of the mathematical expectation states that if there is an X and Y, then the product of those two random variables are equal to the product of the mathematical expectation of the individual random variables. In other words,
E(XY)= E(X) E(Y), provided all the expectations exist.
This property of the mathematical expectation also has the generalized form, which states that the product of the ‘n’ number of random variables is equal to the product of the mathematical expectation of the individual ‘n’ number of the random variables. In other words,
E(X1X2 …… Xn) = E(X1)E(X2) …… E(Xn), provided all the expectations exist.
The third property of the mathematical expectation states that if X is some random variable and ‘a’ is some constant, then the product of the constant and the function of that random variable is equal to the product of the constant and the mathematical expectation of the function of that random variable. Also, the sum of the constant and the function of that random variable is equal to the sum of the mathematical expectation of the function of that random variable and the constant. In other words,
E(a f(x))=aE(f(x) and E(a+f(x))=a+E(f(x)), provided all the expectations exist.
The fourth property of the mathematical expectation states that if X is some random variable and ‘a’ and ‘b’ are some constants, then the sum of the product of the constant ‘a’ and the random variable and the other constant ‘b’ is equal to the sum between the product of the constant ‘a’ and the mathematical expectation of that random variable and the other constant ‘b.’ In other words,
E(aX+b)=aE(X)+b, provided that all the expectations exist.
The fifth property of the mathematical expectation states that if X1, X2, …. Xn are ‘n’ random variables and a1,a2,…. ,an are ‘n’ constants, then the following is true:
E(ai xi)= ai E(Xi), provided all the expectations exist.