Linear Functions of Independent Random Variables: What's the value of b? Suppose that $X_1,X_2,\ldots,X_{25}$ are independent random variables from N$(1, 4)$. 
$Y = aX + b$, where $a$ ($a > 0$) and $b$ are constants. $X = \frac{1}{25} \sum_{i=1}^{25} X_i$. We know that $Y$ is a standard normal random variable.
What's the value of $b$? 

Recall that $X_i \sim N(1,4)$, gives $\sum_i X_i \sim N(25, 100)$, hence $X = \frac 1{25}\sum_i X_i \sim N(1,\frac{4}{25})$. This implies that $aX \sim N(a,\frac{4}{25}a^2)$ and $aX + b \sim N(a+b, \frac{4}{25}a^2)$. If $aX+ b \sim N(0,1)$, this implies 
$$ \frac{4}{25}a^2 = 1, \qquad a +b = 0$$
as $a > 0$, $\frac{4}{25}a^2 = 1$ gives $a = \frac 52$, hence $b = -a = -\frac 52$.
