If Z= aX+b with two contants a,b belongs to real numbers and E[Z]=0, V(Z) =1, find a and b.

Question

In this question I tried the approach as
since,
E[Z]=0
E[Z]=a*E[X]+b
E[X]= -b/a

   V[Z]=1  
   V(Z)=a^2*V(X)  
   V(X)=1/a^2  

I am stuck after this.

Answer 1

Solving for $a$,$$a=\pm\frac1{\sqrt{V(X)}}$$

then substitute to the first condition that you found. $$b = -aE[X]=\mp \frac{E[X]}{\sqrt{V(X)}}$$

I think the purpose of the exerise is to construct that

$$\pm\frac{X-E(X)}{\sqrt{V(X)}}$$ has mean $0$ and variance $1$.