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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?

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Haha! I couldn't find anything because my question was that elementary. Thanks to both of you. I wish I didn't have to choose who got the rep points. – MackTuesday Feb 2 '14 at 21:22
up vote 5 down vote accepted

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 $\sum_1^n a_i^2 \sigma_i^2$.

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V.X will be Normal, as linear combination of normal rvs is again normal. Check this.

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