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?

  • $\begingroup$ Potential typo, but $p(1,1) = 0.05$ perhaps? $\endgroup$
    – WaveX
    Oct 8 '20 at 1:32
  • $\begingroup$ @WaveX definitely, it's the only one that fits and is off by just an extra zero... $\endgroup$
    – gt6989b
    Oct 8 '20 at 1:48

Just apply the definition of conditional probability.   You have already used the law of total probability to evaluate the marginal probabilities.

$$\begin{align}P(X=2\mid Y=2)&=\dfrac{P(X=2\cap Y=2)}{P(Y=2)}\\[2ex]&=\dfrac{p(2,2)}{p(0,2)+p(1,2)+p(2,2)}\end{align}$$

And how do I know if X and Y independent random variables?

Two random variables are defined as independent when their joint probability function equals the product of their marginal probability functions, for every pair of values in their joint support.

$$X\perp Y\iff \forall x{\in}\{0,1,2\}~\forall y{\in}\{0,1,2\}~\bigl(p(x,y)=p_{\small X}(x)\cdot p_{\small Y}(y)\bigr)$$

So, does this hold?


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