Conditional expectation of uniformly distributed variable

Hint: Let $W=Z-\alpha=\beta X+Y$. Since $\alpha$ is a constant, $\mathbb{E}\left[X\mid Z\right]=\mathbb{E}\left[X\mid W\right]$.

  1. Compute the CDF of $X$ and $W$: $$ F_{X,W}(x,w) =\mathbb{P}(X\leq x,W\leq w) $$
  2. Compute the joint PDF of $X$ and $W$ by taking derivatives: $$ f_{X,W} =\frac{\partial^{2}}{\partial x\partial w}\left[F_{X,W}\right] $$
  3. Compute the density of $X$ conditional on $W$: $$ f_{X\mid W}(x\mid w)=\frac{f_{X,W}(x,w)}{f_{X}(x)} $$
  4. Use the conditional density to resolve the expectation: $$ \mathbb{E}\left[X\mid W\right] =\int x f_{X\mid W}(x\mid w)dx $$

Tags:

Uniform Distribution

Conditional Expectation

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