probability or statistics - TransformedDistribution with Conditioned

Is the following attempt beyond Mathematica 11?

Z = TransformedDistribution[ (A + B)/2 \[Conditioned] A < B, {A \[Distributed] NormalDistribution[mA , sA], B \[Distributed] NormalDistribution[mB , sB]}]

When I try to get Mathematica to show me the PDF of Z, it doesn't work. I tried:

 PDF[Z, y]

Answer

It is possible to derive an exact solution to this problem.

Given: $X$ and $Y$ are independent random variables where $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$, with parameter conditions:

Problem: Find the pdf of $\frac{X+Y}{2} \; \big| \; X < Y$

1. Joint pdf of $(X,Y)$:

By independence, the joint pdf of $(X,Y)$, say $f(x,y)$ is simply the product of the individual pdf's:

2. Let $V = X - Y$. Then $V \sim N(\mu_1 - \mu_2, \sigma_1^2 + \sigma_2^2)$ with cdf $\Phi(v)$.

Let constant $c = P(X which is: (take care here with non-standard Mma notation)

3. Conditional joint pdf:

The conditional pdf $f\big((x,y) \; \big| \; X is then fcon:

where all the dependence is captured within the fcon statement using the Boole statement, and we can enter the 'domain' as a rectangular structure on the real line, i.e.

domain[fcon] = domain[f]

4. Transformation $Z = \frac{X+Y}{2}$

Given the conditional joint pdf $f\big((x,y) \; \big| \; X ... let $Z = \frac{X+Y}{2}$ and $W = X$. Then the joint conditional pdf of $(Z,W)$, say $g(z,w)$, is obtained with:

where I am using the Transform function from the mathStatica package for Mathematica, and the domain can again be entered as a rectangular set as:

Then, the marginal pdf of $Z = \frac{X+Y}{2}$ is:

... which is the exact solution. All done.

The following plot compares:

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  the exact symbolic pdf derived above (red dashed curve)

  
  


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  ... to the Monte Carlo simulated pdf (squiggly blue curve)

  
  

... here when: $\mu_1 = -1, \mu_2 = 4, \sigma_1 = 1, \sigma_2 = 12$

Looks fine.