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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 - 2X + )
2
2
2
2
= E(X ) - 2E(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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