Doubt about conditional probability

Question

I have a doubt about conditional probability.

Suppose I know $P^X, P^Y$ that are (fixed) distribution function of random variables $X, Y$.

To which conditions is $P(X|Y$) restricted to?

Example

$X \sim Binomial(10, \frac 13)$, $Y \sim Binomial(12, \frac 12)$.

Let $P(X = k | Y = m) = f(k, m)$.

I cannot choose $f(k, m)$ freely. For example, if I set $f(k, k) = 1 \ \ \forall \ k$, then this implies $X = Y$ (doesn't it?) But then again $X$ and $Y$ have different distribution function so it cannot be that $X=Y$.

I'm a little confuse in this regard, please help me understand better!

Answer 1

The constraints on $f$ defined by $f(k,m)=P(X=k|Y=m)$ in terms of $p$ and $q$ defined by $p(k)=P(X=k)$ and $q(m)=P(Y=m)$ are that, for every $(k,m)$, $$f(k,m)\geqslant0,\qquad\sum_jf(k,j)q(j)=p(k),\qquad\sum_if(i,m)=1.$$ The set $C(p,q)$ of solutions $f$ is convex and non empty since $f:(k,m)\mapsto p(k)$ is always a solution.

The determination of $C(p,q)$ for some given distributions $p$ and $q$ is a nontrivial task, which is equivalent, at least when $q(m)\gt0$ for every $m$, to the determination of the couplings of $p$ and $q$. These are the probability measures $c$ such that, for every $(k,m)$, $$\sum_jc(k,j)=p(k),\qquad\sum_ic(i,m)=q(m).$$ Obviously, the passage from $c$ to $f$ reads $$c(k,m)=f(k,m)q(m).$$ Thus, perusing some classical references on the subject of coupling seems mandatory to anybody interested in this question.

Answer 2

The conditional distribution of $X$ given $Y=y$ is, by definition, the function $$ \mathcal{B}(\mathbb{R})\times\mathbb{R}\ni (A,y)\mapsto \pi(A\mid y) $$ satisfying the following three properties: 1) $y\mapsto \pi(A\mid y)$ is Borel measurable for all $A\in\mathcal{B}(\mathbb{R})$, 2) $A\mapsto \pi(A\mid y)$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ for all $y\in\mathbb{R}$, and 3) $$ P(X\in A,Y\in B)=\int_B \pi(A\mid y) P_Y(\mathrm dy) $$ for all $A,B\in\mathcal{B}(\mathbb{R})$. These three properties uniquely determines the conditional distribution which is often denoted by $P(X\in \cdot\mid Y=\cdot)$.

Exercise: Show that when $Y$ is discrete then $$ \pi(A\mid y)= \begin{cases} P(X\in A\mid Y=y),\quad &\text{if }\, P(Y=y)>0,\\ \mathbf{1}_A(0),&\text{if }\,P(Y=y)=0 \end{cases} $$ is the conditional distribution of $X$ given $Y=y$.