# #StackBounty: #self-study #bayesian #continuous-data #uniform #analytical Two dependent uniformly distributed continuous variables and …

### Bounty: 50

I am trying to solve the following exercise from Judea Pearl’s Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference.

2.2. A billiard table has unit length, measured from left to right. A ball is rolled on this table, and when it stops, a partition is placed at its stopping position, a distance $$x$$ from the left end of the table. A second ball is now rolled between the left end of the table and the partition, and its stopping position, $$y$$, is measured.

a. Answer qualitatively: How does knowledge of $$y$$ affect our belief about $$x$$? Is $$x$$ more likely to be near $$y$$ , far from $$y$$, or near the midpoint between $$y$$ and 1?

b. Justify your answer for (a) by quantitative analysis. Assume the stopping position is uniformly distributed over the feasible range.

For b., I clearly need to use Bayes’ theorem:

$$P(X|Y) = dfrac{P(Y|X)P(X)}{P(Y)}$$

where I expressed

$$P(X) sim U[0,1] = begin{cases} 1, text{where } 0 leq x leq 1\ 0, text{else} end{cases} \ P(Y|X) sim U[0,x] = begin{cases} 1/x, text{where } 0 leq y leq x\ 0, text{else} end{cases}$$

I tried getting $$P(Y)$$ by integrating the numerator over $$X$$.

$$int_{-infty}^{infty} P(Y|X)P(X)dx = int_{0}^{1}P(Y|X)cdot 1 dx = int_{0}^{1}dfrac{1}{x} dx$$

But the integral doesn’t converge.

I also tried to figure out the numerator itself, but I don’t see how $$frac{1}{x}$$ can represent $$P(X|Y)$$.

Where did I go wrong?

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