In: Statistics and Probability
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).
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