Study related to concepts of Probability and Statistics in Math is based on the fact that in a Probability distribution we determine either of the following:
Addition Theorem of ExpectationsBack to Top
E (b) = ∑ r i =1* bi * F (bi) = b1 F (b1) + b2 F (b2) +……..+ bn F (br).
Similarly, mathematical expectation values can also be evaluated for other discrete or continuous random variables. But what method we must follow in order to add mathematical expectations of two random variables? For this purpose we use Addition Theorem of expectations. If we add two random variable values, then what will be the expected value for the resulting variable? This can be expressed as summation of Mathematical Expectation value of both variable with a necessary condition that both variables must be existent.
Suppose we have two random variables “a” and “b” such that their mean values are given as E (a) and E (b) respectively. Then value of expected value according to addition postulate of expectations can be given as follows:
E [a + b] = E [a] + E [b],
In a more comprehensive way we can say that mathematical expectation for summation of a finite Set of variable: a, b,……,z with their expected values as: E [a], E [b] …….E [z] can be given as follows:
E [a + b + c + d…..+ z] = E [a] + E [b] + E [c] + E [d] + …….. + E [z],
So, general explanation for addition theorem states that expected values of sum of 'n' variables can be calculated as sum of their individual mathematical expectation values.
Multiplication Theorem of ExpectationsBack to Top
E(y) = ∑ r i =1* yi * H (yi) = y1 H (y1) + y2 H (y2) +……..+ yn H (yr).
Likewise, mean values for other variables can also be found. If we multiply two or more random variables then what should be the mathematical expected value for resulting expression? For this we use multiplication theorem of expectations. If we multiply 2 random variables (discrete or continuous variables) say “x” and “y” such that their expected values are specified as E (x) and E (y) respectively. Then value of Mathematical Expectation of product according to multiplication method of expectations can be evaluated as follows:
E [x * y] = E [x] + E [y],
In a more generalized way we can say that value of expected value for product of a fixed Set of “k” variables: x, y,……,z with their Mean values as: E [x], E [y] …….E [z] can be calculated as follows:
E [x * y * u * v * w * ….. z] = E [x] * E [y] * E [u] * E [v] * E [w]……..* E [z],
So common way to state the multiplication theorem of expectations is to take product of 'k' variable and show the expected value of resultant as the product of their individual mathematical expectation values.