- #1
Bassalisk
- 947
- 2
Hello,
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
I understand everything up to the part where the final function is drawn.
[PLAIN]http://pokit.org/get/b750c434c9d1e0ba259da0972158b514.jpg
This is pη 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?
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
I understand everything up to the part where the final function is drawn.
[PLAIN]http://pokit.org/get/b750c434c9d1e0ba259da0972158b514.jpg
This is pη 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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