# Homework Help: Is it right to condition a RV on dependent RV?

1. Oct 23, 2012

### devonho

1. The problem statement, all variables and given/known data

Let there be two discrete random variables:

$X \in \lbrace 1,2,3,4,5,6,7,8,9,10 \rbrace \quad \text{where } P[X] \text{ is uniformly distributed over the sample space of } X \text{.}$

$B = \left\lbrace \begin{array}{cl} 1 & \text{if} \quad X>4 \\ 0 & \text{otherwise}\\ \end{array}\right.$

$P[B \mid X]=\left\lbrace \begin{array}{cl} 0 & \text{if} \quad x \in \lbrace 1,2,3,4 \rbrace\\ 1 & \text{if} \quad x \in \lbrace 5,6,7,8,9,10 \rbrace\\ \end{array}\right.$

$P[X] = {1\over10}$

$P[X,B] = P[B \mid X]P[X] = \left\lbrace \begin{array}{cl} 0 &\text{if} \quad x \in \lbrace 1,2,3,4 \rbrace\\ {1\over 10} & \text{if} \quad x \in \lbrace 5,6,7,8,9,10 \rbrace\\ \end{array}\right.$
The above should be agreeable. But what about:

$P[X,B] = P[X \mid B]P$

Since B is dependent on X, is it meaningful or even correct to write an expression for P[X|B]?

2. Relevant equations

3. The attempt at a solution
I think no because the conditional probability will then be recursive.

2. Oct 24, 2012

### haruspex

It's perfectly ok to discuss the probability of an underlying event given an observation concerning it.
P[B=0] = 0.4
P[X=1|B=0] = 1/4
P[X=1 & B=0] = 0.1
etc.