Expectation of Conditional Joint Density

Question

I am working on a problem and am really confused.

Problem: Find E(X|Y) with

P(X,Y) = (1,1) = .4

P(X,Y) = (1,0) = .5

P(X,Y) = (0,0) = .1

What I have done so far:

I drew a x y graph (plotting the points above) which makes a triangle

Written P(X|Y=0) and P(X|Y=1) but is this on the right track? Do we need to use the conditionals that I wrote above? Is there a formula we can use to solve for this expectation?

I believe that based on the graph, y can take values 0 (.6) and 1 (.4) while x can take values 0 (.1) and 1 (.9)

Answer 1

Apply the usual definition of expectation, but using conditional probability measures.

$$\begin{split}\mathsf E(X\mid Y{=}0)&=0\cdot\mathsf P(X{=}0\mid Y{=}0)+1\cdot\mathsf P(X{=}1\mid Y{=}0)\\[1ex]&=0+\frac {\mathsf P(X{=}1,Y{=}0)}{\mathsf P(Y{=}0)}\\[1ex]&=\dfrac{0.5}{0.1+0.5}\\[1ex]&=\frac 5 6\\[1ex]&= 0.8\dot{\overline 3}\end{split}$$