Plotting three related distributions on the same axis

Question

The following solution to this problem was found by @Wolfies:

(E^(-((-2 z + \[Mu]1 + \[Mu]2)^2/(2 (\[Sigma]1^2 + \[Sigma]2^2)))) Sqrt[2/\[Pi]] (-1 + Erf[((z - \[Mu]2) \[Sigma]1^2 + (-z + \[Mu]1) \[Sigma]2^2)/(Sqrt[2] \[Sigma]1 \[Sigma]2 Sqrt[\[Sigma]1^2 + [Sigma]2^2])]))/(-1 + Erf[(\[Mu]1 - \[Mu]2)/(Sqrt[2] Sqrt[\[Sigma]1^2 + \[Sigma]2^2])] Sqrt[\[Sigma]1^2 +[Sigma]2^2])

Note that $Z = \frac{X + Y}{2}$ where $X$ and $Y$ are independent random variables that both follow the Normal distribution, with possibly different means and variances (denoted by subscripts $1$ for $X$ and $2$ for $Y$).

I am now trying to plot the two normal distributions together with the random variable $Z$ described by Wolfies' solution, on the same axis. What gets me is that $Z$ does not have the size I would expect relative the two Normal distributions; the distribution for $Z$ is too large.

Below is a plot of $Z$ and $X$ on two different axis, and the size problem is clear:

Answer 1

After correcting the density you should get something like the following:

Manipulate[Plot[{PDF[NormalDistribution[μ1, σ1], z],
   PDF[NormalDistribution[μ2, σ2], z],
   (E^(-((-2 z + μ1 + μ2)^2/(2 (σ1^2 + σ2^2)))) Sqrt[2/π]*
      (-1 + Erf[((z - μ2) σ1^2 + (-z + μ1) σ2^2)/(Sqrt[2] σ1 σ2 Sqrt[σ1^2 + σ2^2])]))
    ((-1 + Erf[(μ1 - μ2)/(Sqrt[2] Sqrt[σ1^2 + σ2^2])]) Sqrt[σ1^2 + σ2^2])}, {z, -10, 10},
  PlotLegends -> {"\!\(\*SubscriptBox[\(X\), \(1\)]\)", 
    "\!\(\*SubscriptBox[\(X\), \(2\)]\)", 
    "(\!\(\*SubscriptBox[\(X\), \(1\)]\)+\!\(\*SubscriptBox[\(X\), \
\(2\)]\))/2 given \!\(\*SubscriptBox[\(X\), \(1\)]\) < \
\!\(\*SubscriptBox[\(X\), \(2\)]\)"},
  PlotRange -> All],
 {{σ1, 1, "\!\(\*SubscriptBox[\(σ\), \(1\)]\)"}, 0.1, 5, Appearance -> "Labeled"},
 {{σ2, 1, "\!\(\*SubscriptBox[\(σ\), \(2\)]\)"}, 0.1, 5, Appearance -> "Labeled"},
 {{μ1, 0, "\!\(\*SubscriptBox[\(μ\), \(1\)]\)"}, 0, 5, Appearance -> "Labeled"},
 {{μ2, 0, "\!\(\*SubscriptBox[\(μ\), \(2\)]\)"}, 0, 5, Appearance -> "Labeled"},
 TrackedSymbols :> {μ1, μ2, σ1, σ2}]