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Mathematical Expectation

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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:

1. Probability of occurrence of all arbitrary values of a Random Variable (when the variable is discrete) or,

2. The probability of occurrence of some value that is supposed to lie in a specific interval.

We discuss about two different kinds of random variables in statistical study. One is said to be Discrete Random Variable while the other one is known as continuous random variables. Formulas or method we implement in calculation of statistical data are supposed to vary according to the type of variable we introduce in the problem. Mathematical Expectations (also known as Mean values of random variables is indicated by means of E [X].

When variable we consider as discrete value, then value of mathematical expectation is evaluated as summation of the product of the arbitrary value of the random variable and their corresponding probability mass function value. In case the random variable we choose is of continuous behavior, then value of mathematical expectation is calculated as Integration of multiplication of arbitrary values of random variable and probability density function of those random variables. Remember that probability function we use in case of a continuous variable is different from that we used for discrete valued variable when used in mathematical expectation problems. For instance, let us consider a random variable “x” which takes the following arbitrary values: a1, a2…an. Probability values for respective values are as follows:

F (a1), F (a2)….F (an),

Value of mathematical expectation, also named as expected value is given as:

E(x) = ∑n i =1 * i F (xi) = a1 F (a1) + a2 F (a2) +……..+ an F (an).

Addition Theorem of Expectations

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Mathematical expectation or Mean or expected value for a Random Variable “b” that can have following arbitrary values: b1, b2…br with their respective values of Probabilities as: F (b1), F (b2)….F (br) can be evaluated as follows:

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 Expectations

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Expected value is calculated for summation of two or more random variables using addition postulate, in the same way expected value for product of variables is calculated using multiplication postulate. Suppose we have a variable “y” which can take following random values: y1, y2…yr with their individual Probability values as: H (y1), H (y2)….H (yr), and then its expected value can be given as:

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.