Question

In: Statistics and Probability

Let X have mean ? and variance ?^2. (a) What are the mean and variance of...

Let X have mean ? and variance ?^2. (a) What are the mean and variance of -X? Explain intuitively. (b) Find constants a and b such that the random variable Y=aX +b has mean 0 and variance 1 (this is called standardization).

Solutions

Expert Solution

Solution:

We are given that: X has Mean = and Variance = .

Part a) What are the mean and variance of -X?

Mean has effect of change of origin as well as change of scale.

That is: E(aX + b) = a E(X) +b

Variance has effect of change of scale but not the change of origin.

That is : Var(aX+b) = a2 Var(X) + Var(b) = a2 Var(X) + 0 = a2 Var(X)

Since variance of constant is 0.

Thus the mean and variance of -X are:

Mean of -X :

E(-X) = - E(X)

E(-X) =

Variance of - X:

Part b) Find constants a and b such that the random variable Y=aX +b has mean 0 and variance 1

We want find a and b such that Mean is 0 and variance is 1.

Thus :

E(Y) = 0

E(aX + b) = 0

a E(X) + b = 0

      .........(1)

Now consider:

Var(Y) = 1

Var(aX + b) = 1

a2 Var(X) + Var(b) = 1

a2 Var(X) +0 = 1

a2 Var(X) = 1

     .......(2)

Substituting (2) in equation (1) , we get :

Thus constants a and b are:

and


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