Calculating CDF of summation of normal variable

Question

I have iid normal random variables $X_i$ from i = 1,..n. And I want to calculate pdf of:

$ \sum_{i=1}^{n} a_i x_i $

So I know that I should be calculating the MGF and then get the product of those variables. But I could not calculate the CDF of this normal distribution. How can I calculate the CDF of this normal distribution by using the properties of Moment Generating Function ?

Answer 1

If your $X_i\sim N(\mu;\sigma^2)$ the pdf of your sum is

$$Y\sim N\left(\mu\Sigma_i a_i;\sigma^2\Sigma_i a_i^2 \right)$$

But the CDF of a Gaussian cannot be expressed by elementary function.

$$F_Y(y)=\int_{-\infty}^y f_Y(t)dt$$

This integral can only be written in terms of the special function erf (error function)