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  • $\begingroup$ Check TransformedDistribution[ 1/2 x + 1/2 y, {x \[Distributed] NormalDistribution[\[Mu], \[Sigma]], y \[Distributed] NormalDistribution[\[Mu], \[Sigma]]}] and see if you agree with its output. $\endgroup$ Commented Aug 21, 2018 at 8:38
  • $\begingroup$ @b.gatesucks Thanks for your comment. TransformedDistribution yields practically the same result as my approach. $\endgroup$ Commented Aug 21, 2018 at 8:56
  • $\begingroup$ It would help if you were more specific with your terms. Are you interested in the linear combination of two normally distributed random variables, say, $Z=X/2+Y/2$? Or do are you interested in a mixture distribution where there are 3 random variables: $Z=\alpha X+(1-\alpha)Y$ where $X$ and $Y$ are normally distributed random variables (possibly independent of each other) and another Bernoulli random variable $\alpha$ that has $Pr(\alpha=1)=p$ and $Pr(\alpha=0)=1-p$ ? $\endgroup$ Commented Aug 21, 2018 at 21:07