Consider the random variable $y_M$ which equals the sum of $M$ independently uniformly distributed random variables; i.e., $y_M = x_1 + x_2 +\ldots + x_M$, with the assumption that each of the $x_i$ are independently and uniformly distributed. The variance of the sum equals the sum of the variances of each of the $x_i$. That is, $Var[x_1 + x_2 + \ldots + x_M]$ $= Var[x_1] + Var[x_2] + \ldots + Var[x_M]$. Likewise, $Mean[x_1 + x_2 + \ldots + x_M]$ $= Mean[x_1] + Mean[x_2] + \ldots + Mean[x_M]$.
Given this I need to do the following:
- Construct $Y(x_1, x_2 , \ldots, x_M) = a(x_1 + x_2 + \ldots + x_M) + b$.
- Find $a$ (slope) and $b$ (intercept) such that $Y$ has zero mean and unit variance.
This seems to be a transformation technique (of which I'm unaware). Any thoughts on how to approach this problem?