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I have a really tough time understanding this concept although this isn't anything more complex than a composition of functions.

I have a done example from my book that I am trying to interpret.

[itex] p_\xi (x)=\frac{1}{\pi} [/itex]

for an interval between [0,∏]

The transformation is given by law:

[itex] \eta=sin(\xi) [/itex] interval from [0,∏]

Find the PDF for η.

Now I understand that sin(x) isn't monotonous on whole interval.

So we have to divide this into 2 monotonous functions.

From [0,pi/2] and from [pi/2, pi]

And we use formula for that monotonous transformation 2 times.

[PLAIN]http://pokit.org/get/d635e99a44c3c266a5c93e734d61af32.jpg [Broken]

I understand everything up to the part where the final function is drawn.

[PLAIN]http://pokit.org/get/b750c434c9d1e0ba259da0972158b514.jpg [Broken]

This is p_{η}graph which is given in my book.

But I cannot wrap my head around it.

First question:

How can we just add those 2 monotonous transformations, aren't we losing pieces of information?

And where is that second part, where transformation is monotonously decreasing?

Any help with my intuition here?

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# Transformation of random variable

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