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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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  • $\begingroup$ 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. $\endgroup$ – Minus One-Twelfth Mar 21 at 21:37
  • $\begingroup$ Yes they are independent, thanks. $\endgroup$ – holly Mar 21 at 21:42
  • $\begingroup$ If I use this rule, then is $X-Y\sim \cal N(0,2\sigma^2)$? $\endgroup$ – holly Mar 21 at 21:48
  • $\begingroup$ Yes, that is correct. $\endgroup$ – Minus One-Twelfth Mar 21 at 21:52
  • $\begingroup$ Thank you very much! Greetings Holly $\endgroup$ – holly Mar 21 at 21:53