# Distribution function of X-Y for normally distributed random variables [duplicate]

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I have two independent normally distributed random variables $$X,Y\sim \cal N(\mu,\sigma^2)$$ and want to calculate the distribution of $$X-Y$$. I tried with $$F(z)=P(X-Y \leq z)$$ but failed. Does anyone has an idea how I approach it?

## marked as duplicate by Mike Earnest, Community♦Mar 21 at 21:51

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• Are $X$ and $Y$ independent? If so, do you know a general result about the distribution of a linear combination of independent normal random variables? You can see here for example: newonlinecourses.science.psu.edu/stat414/node/172. – Minus One-Twelfth Mar 21 at 21:37
• Yes they are independent, thanks. – holly Mar 21 at 21:42
• If I use this rule, then is $X-Y\sim \cal N(0,2\sigma^2)$? – holly Mar 21 at 21:48
• Yes, that is correct. – Minus One-Twelfth Mar 21 at 21:52
• Thank you very much! Greetings Holly – holly Mar 21 at 21:53