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Let $x \sim U[0,1]$ and $y\sim U[0,1]$. Let $z= \omega\, x+ (1-\omega)\,y$, where $\omega\in[0,1]$. The pdf of $z$ is a trapezoidal distribution over $[0,1]$: \begin{equation*} \begin{aligned} f(z)&= \left\{\begin{array}{cl} \dfrac{z}{\omega (1-\omega)} &\text{if}\,\,\,z \in[0,\min\{\omega, 1-\omega\}] \\[0.08in] \dfrac{1}{\max\{\omega, 1-\omega\}} &\text{if}\,\,\,z \in(\min\{\omega, 1-\omega\},\max\{\omega, 1-\omega\}] \\[0.08in] \dfrac{1-z}{\omega (1-\omega)} & \text{if}\,\,\, z \in(\max\{\omega, 1-\omega\},1]. \end{array} \right. \end{aligned} \end{equation*} I am searching for the conditional pdfs $f(z|x)$ and $f(x|z)$ (or the joint $f(z,x)$). How should I proceed?

Philipponat
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  • See https://stats.stackexchange.com/questions/243887, https://stats.stackexchange.com/questions/190469, https://stats.stackexchange.com/questions/31804/. – whuber Nov 18 '23 at 16:20
  • @whuber These are about the sum of uniforms, I am interested in the weighted average. – Philipponat Nov 18 '23 at 16:31
  • The weighted average is a sum of uniforms having potentially different upper limits, that's all. For instance, when $X\sim U(0,1)$ and $\omega\gt 0$ then $\omega X \sim U(0,\omega).$ – whuber Nov 18 '23 at 16:35
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    @whuber Thank you that was helpful. – Philipponat Nov 21 '23 at 14:01
  • Concerning the new question, you can follow the procedures given at https://stats.stackexchange.com/questions/232085/ or https://stats.stackexchange.com/questions/210394 which concern a slightly more complicated version of your question. The thread at https://stats.stackexchange.com/questions/584874 also is related (with uniform distributions replaced by exponential distributions). – whuber Nov 21 '23 at 15:19

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