Finding Conditional Distributions

Question

I am working on a problem and am a bit stuck.

The problem: Find conditional distribution and expectation of X given Y

(Sorry I did not know how to format into a table)

\begin{array}{l:l|c}X & Y & P \\\hline 0 & 0 & .1 \\ 1 & 0 & .2 \\ 2 & 0 & .1 \\ 0 & 1 & .4 \\ 1 & 1 & .2 \end{array}

What I have done so far:

For finding E(X|Y) split into:

E(X|Y=0) = $(0)(.1)+(1)(.2)+(2)(.1)\over(.1+.2+.1)$ = $.4\over.4$ = 1

E(X|Y=1) = $(0)(.4)+(1)(.2)\over(.4+.2)$ = $.2\over.6$ = $1\over3$

This doesn't exactly seem right however. For creating the conditional distribution do we just include these values? Is this the conditional distribution?

Answer 1

This doesn't exactly seem right however.

That is correct. Thos are the conditional distributions for $X$ when given a value for $Y$.

You may wish to express it as $\mathsf E(X\mid Y)=\mathbf 1_{Y=0}+\tfrac 13\mathbf 1_{Y=1}$

For creating the conditional distribution do we just include these values? Is this the conditional distribution?

Similar. Evaluate the Conditional Probability Mass Functions for $X$ when given values for $Y$.

Eg $\mathsf P(X=0\mid Y=0)=\dfrac{0.1}{0.1+0.2+0.1}=\dfrac 14$, and similarly $\mathsf P(X=1\mid Y=0)=\dfrac 12$, etcetera.