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Properties of expectation

1)      Expectation of constant is a constant itself.

Proof)

Let us say that a is the constant and f(x_i) is the probability function

When i = 1,2,3……. n therefore

E (a) =  a f(x₁) + a f(x₂) ………..a f(x_n)

     = a[ f(x₁) + f(x₂) + ……..+ f(x_n) ]

    = a ∑ f(x_i)                                                                                      

   E(a)= a(1) = a

 E(a) =   a

2)     E (ax + b) = a E (X) + b

If (ax + b) is a r.v then f(xi) is a probability function where i = 1, 2 ,3……n

E ( aX + b) =

                    =

                    =