Let $X$ be a random variable geometrically distributed with parameter $p$ and let $Y$ be a random variable with Poisson distribution following parameter $p$. $X$ and $Y$ are independent.
Calculate $E[X | Y]$.
Calculate $E[|X - Y|]$.
How does one approach these calculations?
For the first one we know by definition that $$E[Y|X] = \sum_y yf_{Y|X}(y|X)$$ but we can write $$f_{Y|X}(y|x) = \frac{f_{Y,X}(y,x)}{f_X(x)}$$ and by independence $$f_{Y|X}(y|x) = \frac{f_{Y}(y)f_X(x)}{f_X(x)}$$ so we just have $$\sum_y yf_Y(y)=E(Y)$$
For the second you can do a couple of things. Personally, my plan of attack would be to find the density of $Z=Y-X$ by transform of random variable method (or discrete convolution formula), then find the distribution of $|Z|$ and take the expected value.
Edit: that second answer was a cop out. Lets look at $Z=Y-X$, we know from the convolution formula $$F_{X-Y}(z)= P(X-Y <z)$$ $$\int_{-\infty}^{\infty}F_y(z+x)p_X(x)dx$$ Okay so we can evaluate that integral and find the density by differentiating the result, or we can push the derivative into the integral and write $$\int_{-\infty}^{\infty}p_Y(z+x)p_X(x)dx$$ Okay so you can evaluate that expression and get some density using the definition of poisson/geometric. Lets find the final transform we need by defining the resulting density to map to the random variable $K$
So we know how $K$ is distributed from the previous step. Lets find the transform of $O=|K|$ How do we find this? Well we know $$P(O<o)=P(|K|<o)=P(-o < K < o)= P(K < o)-P(-o < K)=P(K<o)-P(o>K)$$
Which we can compute using the CDF we found above.