Sorry if this is a basic question. I don't know much about statistics and the closest thing I found involved unit vectors, a case I don't think is easily generalizable to this problem.
I have a reference vector $\mathbf V$ in some $\mathbb R^n$.
I have another vector in $\mathbb R^n$ of independent random variables, each with Gaussian distribution, each with the same standard deviation $\sigma$. Let's call the vector $\mathbf X$.
What is the probability distribution of $\mathbf V\cdot \mathbf X$?
Surely this is a famous problem with a widely known solution. Or is there an elegant approach to the problem?
Best Answer
You are asking for the distribution of $Y=a_1X_1+\cdots +a_n X_n$, where more generally the $X_i$ are independent normal, means $\mu_i$, variances $\sigma_i^2$. The random variable $Y$ has normal distribution, mean $\sum_1^n a_i \mu_i$, variance $\sum_1^n a_i^2 \sigma_i^2$.