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Statistics

Notes Proof.

2
We can write Var(aX) = E[aX - E(aX)] = E[aX - aE(X)] 2
2
= a E[X - E(X)] = a Var(X).
2
2
Combining the results of theorems 2 and 3, we can write
Var(aX + b) = a Var(X).
2
This result shows that the variance is independent of change origin but not of change of scale.
Remarks:
1. On the basis of the theorems on expectation and variance, we can say that if X is a random
variable, then its linear combination, aX + b, is also a random variable with mean
2
aE(X) + b and Variance equal to a Var(X).
2. The above theorems can also be proved for a continuous random variable.

Example 4: Compute mean and variance of the probability distributions obtained in
examples 1, 2 and 3.
Solution.
(a) The probability distribution of X in example 1 was obtained as

X 0 1 2 3

1 3 3 1
( )
p X
8 8 8 8
From the above distribution, we can write

1 3 3 1
E ( ) 0X = ´ + 1 ´ + 2 ´ + 3 ´ = 1.5
8 8 8 8
To find variance of X, we write

2
2
E X
X
Var(X) = E(X ) - [E(X)] , where ( ) = å X p ( ) .
2
2
1 3 3 1
2
Now, ( ) 0E X = ´ + 1´ + 4 ´ + 9´ = 3
8 8 8 8
2
Thus, Var(X) = 3 - (1.5) = 0.75
(b) The probability distribution of X in example 2 was obtained as
X 2 3 4 5 6 7 8 9 10 11 12 Total
1 2 3 4 5 6 5 4 3 2 1
( )
p X 1
36 36 36 36 36 36 36 36 36 36 36

1 2 3 4 5 6
\ E ( ) 2 ´ + 3 ´ + 4 ´ + 5 ´ + 6 ´ + 7 ´
=
X
36 36 36 36 36 36

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