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Unit 10: Approximate Expressions for Expectations and Variance



            Theorem 2.                                                                            Notes

            E(aX) = aE(X), where X is a random variable and a is constant.
            Proof.
            For a discrete random variable X with probability function p(X), we have :

                       E(aX) = aX .p(X ) + aX .p(X ) + ...... + aX .p(X )
                                1   1    2   2        n   n
                                n
                                      ( ) aE X=
                                  a=  å X i .p X i  ( )
                               i= 1
            Combining the results of theorems 1 and 2, we can write
                       E(aX + b) = aE(X) + b
            Remarks: Using the above result, we can write an alternative expression for the variance of X, as
            given below :
                         = E(X - )  = E(X  - 2X +  )
                                 2
                                      2
                        2
                                              2
                           = E(X ) - 2E(X) +   = E(X ) - 2  + 2
                             2
                                         2
                                                   2
                                              2
                                       2
                             2
                                 2
                           = E(X ) -   = E(X ) - [E(X)] 2
                           = Mean of Squares - Square of the Mean
            We note that the above expression is identical to the expression for the variance of a frequency
            distribution.
            10.1.1 Theorems on  Variance
            Theorem 1.
            The variance of a constant is zero.
            Proof.
            Let b be the given constant. We can write the expression for the variance of b as:

                                      2
                                               2
                       Var(b) = E[b - E(b)]  = E[b - b]  = 0.
            Theorem 2.
                       Var(X + b) = Var(X).
            Proof.
                                                2
            We can write Var(X + b) = E[X + b - E(X + b)]  = E[X + b - E(X) - b] 2
                                             2
                                  = E[X - E(X)]  = Var(X)
            Similarly, it can be shown that Var(X - b) = Var(X)
            Remarks: The above theorem shows that variance is independent of change of origin.

            Theorem 3.
                                2
                       Var(aX) = a Var(X)








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