# Thread: Subtracting two normal distributions

1. ## Subtracting two normal distributions

I have two distributions X~N(.65,.0002) and Y~N(.59,.0003) if I want the mean and SD do I just subtract them? and is X-Y also a normal distribution?

2. Hello,

If they're independent, X-Y will indeed be a normal distribution.

Now some properties... :
$\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[Y]$

$Var(aX+bY)=a^2Var(X)+b^2Var(Y)$ if they're independent. If not, you'll have to introduce the covariance (check your lessons)

So Var(X-Y)=Var(X)+Var(Y), if they're independent. Note that the variance is always positive !

3. Yes X and Y are independent. So would it just be $u_{x-y}$ = .06 and $SD_{x-y}$ = |.0002-.0003|? To add more information to this, X is the diameter of a metal pipe and the Y is the diameter of a pipe clamp so the X must fit inside Y. I'm also trying to figure out the probability that the pipe will fit inside the clamp.

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### subtracting two normal distributions

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