We know $X_1 \sim U[0, 1]$ and $X_2 \sim U[0, 2]$. I'm trying to compute $E[X_1^3 | 2X_1 + X_2 = 2]$.

I was given the hint that $X_1 | 2X_1 + X_2 \sim U[0, 1]$, and from here on I can compute $E[X_1^3 | 2X_1 + X_2 = 2] = \frac{1}{4}$. However, I'm not sure how we can prove the hint.


When $2X_1+X_2=2$, we know that $X_1=\frac{2-X_2}{2}$.

We also know that $X_2\sim U[0,2]$, so $2-X_2\sim U[0,2]$, and therefore $\frac{2-X_2}{2}\sim U[0,1]$.

When $2X_1+X_2=2$, we have that $X_1=\frac{2-X_2}{2}\sim U[0,1]$.


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