Simple joint probability distribution

jetlam
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Homework Statement



Let's say there are two machines, X and Y. X is connected to Y.

* If X is turned on, Y turns on 50% of the time.
* If Y turns on (through X being turned on) then it breaks 25% of the time.
* Y won't break spontaneously and it can only be turned on through X.

What is the joint probability distribution between Y turning on and Y breaking?
Considering the set of times that X turns on, what is the mutual information between Y turning on and Y breaking (how many bits)?

Thanks.

2. The attempt at a solution

I'm not really sure how to start or how to make the joint probability distribution.
 
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Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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