# Variance of Gaussian linear combination

I have two independent gaussian distributions to combine and I have a doubt.

Let's say we have $X \sim N(\mu_x,\sigma^2_x)$ and $Y \sim N(\mu_y,\sigma^2_y)$. I want to mix the two variables with a 0.5 weight each, so I obtain a new variable $Z=0.5*X+0.5*Y$ with mean $\mu_z=0.5*\mu_x+0.5*\mu_y$.

Is it right to say that $\sigma^2_z=0.5^2*\sigma^2_x+0.5^2*\sigma^2_y$?

Because I clearly obtain less variance than intuitively expected when combining distributions.

• Linear combination is very different than a mixture. If you are interested in a linear combination, your formula is correct. – JohnK Jan 20 '16 at 14:41
• Sorry, I was refering with mixture as a linear combination with sum of weights=1 (I think someone thought that to me in a course), not as a mixture of random variable. That being said, isn't that strange that the variance reduces? I've never thought of this problem before. – Arpayon Jan 20 '16 at 14:45
• Your intuition is "biased", i.e., wrong exectation. Take the special case of Y having the same mean and variance as X, i.e., X and Y are i.i.d., As you take the average of more and more i.i.d. Gaussian random variables, the variance of that average will decrease. Surely you are familiar with that concept. That's what's happening here. – Mark L. Stone Jan 20 '16 at 14:54
• Hi Mark. In my case the normal variables are truncated expected lifetimes for males and females. i.e. I have 22 years for 60 years old males and 24 years for 60 years old females. Now, if I want to find some sort of general population truncated expected lifetime it's intuitive that's 23 (if I use same weight for male and female populations). Is variance decreasing because the result cannot be outside the previous intervals? Am I guessing it right? – Arpayon Jan 20 '16 at 15:39
• When you average (for instance, 2) independent random variables, an extreme value in one variable tends not to be matched with a similar direction extreme value in the other variable, hence making the average less extreme than the individual random variables, hence the decreased variance. – Mark L. Stone Jan 20 '16 at 15:56

Jeremias K's answer shows the formula for the variance of a weighted sum of random variables. In particular, taking the special case of all weights being $1/n$ for a weighted sum of n independent random variables, the variance of the weighted sum equals $1/n^2$ times the sum of the variances of the individual random variables. If the random variables are also identically distributed, then the variance of the weighted sum equals $1/n$ times the variance of an individual random variable.
If $X$ and $Y$ are independent, $Cov(X,Y) = 0$. Since also, $Var(aX+bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)$, it is in fact right to calculate the variance as you did.