Question

In the notation below, X is the random variable, c is a constant, and V refers to the variance.

Which of the following laws of variance is not true?
-V(c) = 0
-V(cX) = c2 V(X)
-V(X + c) = V(X) + c
-None of these choices.

Answer 1

-V(X + c) = V(X) + c

Explanation:

Using the propierties of the variance:

• The variance of a constant is zero:

`Var(k)=0`

Where k is an arbitrary constant. So:

-V(c) = 0 is true.

• If all values are scaled by a constant, the variance is scaled by the square of that constant:

`Var(kX)=k^2V(X)`

Where k is an arbitrary constant. Therefore:

-V(cX) = c2 V(X) is true.

• Variance is invariant with respect to changes in a location parameter. That is, if a constant is added to all values of the variable, the variance is unchanged:

`Var(X+k)=V(X)`

Where k is an arbitrary constant. Hence:

-V(X + c) = V(X) + c is not true.