Question

Let X be a random variable with mean μ and standard deviation σ. Consider a new random variable Z, obtained by subtracting the constant μ from X and dividing the result by the constant σ: Z = (X –μ)/σ. The variable Z is called a standardised random variable. Use the laws of expected value and variance to show the following: a E(Z) = 0 b V (Z) = 1

Answer 1

Given:

                (1)

              (2)

           (3)

To show:

(i)      

and

(ii)     

Proof:

(i)

Taking Expectation on both sides of (3), we get:

Taking out the constant , we get:

i.e.,

    (4)

From (1),

                                (5)

and

since is constant:

                                (6)

Substituting (5) & (6), equation (4) becomes:

Thus, we prove:

(ii)

Taking Variance on both sides of (3), we get:

i.e.,

           

Since and are constants,

Thus, we get:

Using the property of variance, we get:

Applying (2), we get:

This proves: