I have faced the following dilemma while doing some research, so your kind support will be highly appreciated.

I have the following two random variables: $$X_1 \sim N(\mu_1,\sigma^2_1)$$ $$X_2\sim N(\mu_2,\sigma^2_2)$$

and giving $$Z=\frac {X_1 + X_2}{2} $$
I am looking for the joint probability density function $ f(Z,X_1) $

suppose that $\mu_1= \mu_2=a $ and $\sigma^2_1=\sigma^2_2=2a$.
which would give us $Z\sim N(a,a)$

Anyone could help

  • 1
    $\begingroup$ Note that $\begin{pmatrix} Z \\ X_1 \end{pmatrix} = \begin{pmatrix} 0.5 & 0.5 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} X_1 \\ X_2 \end{pmatrix}$. The question is then resolved by the result on the density of a linear transformation of jointly gaussian RVs, which can be seen in any textbook, or in this MSE answer $\endgroup$ – stochasticboy321 Nov 5 '15 at 9:07
  • $\begingroup$ Would you please clarify a little bit more since I am not a mathematician, suppose that $μ_1=μ_2=a $ and $ σ^2_1=σ^2_2=2a $ $\endgroup$ – eMAS Nov 5 '15 at 10:45

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