# Subtracting two normal distributions

• Oct 14th 2009, 12:19 PM
Flipz4226
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?
• Oct 14th 2009, 12:44 PM
Moo
Hello,

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

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

\$\displaystyle 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 !
• Oct 14th 2009, 12:54 PM
Flipz4226
Yes X and Y are independent. So would it just be \$\displaystyle u_{x-y}\$ = .06 and \$\displaystyle 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.