# Simultaneous distribution

We consider two random variables X and Y with simultaneous probability distribution $$p (x, y) = P (X = x ∩ Y = y)$$ given by this table

$$\begin{array}{|c|c|c|c|} \hline x/y& 0 & 1 & 2 \\ \hline 0 & 0.1 & 0.1 & 0.2 \\ \hline 1 & 0.1 & 0.05 & 0.1 \\ \hline 2 & 0.1 & 0.15 & 0.1 \\ \hline \end{array}$$ Question: Find the marginal probability distributions for X and Y. What is P (X = 2 | Y = 2)? Are X and Y independent random variables? Justify the answer.

To find marginal probability distributions

$$p_y(0)\sum _{All\:x}\: p(x,0)=p(0.0)+p(1.0)+p(2.0)=0.1+0.1+0.1=0.3$$ $$p_y(1)\sum _{All\:x}\: p(x,0)=p(0.1)+p(1.1)+p(2.1)=0.1+0.05+0.15=0.3$$ $$p_y(2)\sum _{All\:x}\: p(x,0)=p(0.2)+p(1.2)+p(2.2)=0.2+0.1+0.1=0.4$$

and do the same thing to find x. But I do not understand is what they mean by $$P\left(X=2|Y=2\right)$$. And how do I know if X and Y independent random variables?

• Potential typo, but $p(1,1) = 0.05$ perhaps? Oct 8, 2020 at 1:32
• @WaveX definitely, it's the only one that fits and is off by just an extra zero... Oct 8, 2020 at 1:48

\begin{align}P(X=2\mid Y=2)&=\dfrac{P(X=2\cap Y=2)}{P(Y=2)}\\[2ex]&=\dfrac{p(2,2)}{p(0,2)+p(1,2)+p(2,2)}\end{align}
$$X\perp Y\iff \forall x{\in}\{0,1,2\}~\forall y{\in}\{0,1,2\}~\bigl(p(x,y)=p_{\small X}(x)\cdot p_{\small Y}(y)\bigr)$$