Prove that it is only possible for the expectation of Y to equal the variance of Y if b is less than or equal to 1/4.

Question

A discrete random variable, X, has equal expectation and standard deviation. Y is a transformation of X such that $Y=aX-b$. Prove that it is only possible for the expectation of Y to equal the variance of Y if $b\le1/4$.

So my idea was to say that $E(X)=\sqrt{VAR(X)}$

Also $E(Y)=aE(X)-b$ and $VAR(Y)=a^2VAR(X)$

I equated these expressions to get:

$a^2\sigma^2=a\sigma-b$

Finally, I rearranged this to get:

$a\sigma(1-a\sigma)=b$

However, I am not sure how to proceed. My guess would be I need to use the fact that the standard deviation cannot be negative, but any help would be appreciated. Thanks!st

Answer 1

Hint: consider the function $f(x)=x(1-x)$. What is the range of this function?