Transformation of random variable

[Bassalisk]

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.



$p_\xi (x)=\frac{1}{\pi}$

for an interval between [0,∏]

The transformation is given by law:

$\eta=sin(\xi)$ 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 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?

 

[Stephen Tashi]

This is p_η graph which is given in my book.

If that is supposed to be the graph of a probability density function, I don't understand it either. Is probability plotted on the horizontal axis? And why does the graph have y coordinates as large as 4.5?
.

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

How did the book show the addition? ( The term is "monotonic" , not "monotonous", isn't it?)

When you find the cumulative distribution y = F(x) it's clear you must add the probabilities of two disjoint intervals on the x-axis to get the values of the random variable that are equal or less than y.

 

[Bassalisk]

If that is supposed to be the graph of a probability density function, I don't understand it either. Is probability plotted on the horizontal axis? And why does the graph have y coordinates as large as 4.5?
.


How did the book show the addition? ( The term is "monotonic" , not "monotonous", isn't it?)

When you find the cumulative distribution y = F(x) it's clear you must add the probabilities of two disjoint intervals on the x-axis to get the values of the random variable that are equal or less than y.

This is the final graph. Graph which the problem is asking for. The output function.

This is the law of transformation:
[PLAIN]http://pokit.org/get/76f8d6e186f55a03e47686496639aff5.jpg

ergo the sine wave from 0,pi


and this is the input function:

[PLAIN]http://pokit.org/get/72cd80aa4ae524097162ce20d8ebb105.jpg

ergo the uniformly distributed PDF

 

[Stephen Tashi]

The last two graphs make sense. The graph of $p_{\eta}$ doesn't make sense (to me). Are you taking these figures directly from your course materials?

The figure showing the equation for $p_\eta (y)$ has "$x = f_1^{-1}(x)$" instead of "$x = f_1^{-1}(y)$", so it has errors also.

I think the simplest way to understand why the "transformation" approach works is to first study the "distribution method", which finds the cumulative distribution. Have you studied that method?

 

[Bassalisk]

The last two graphs make sense. The graph of $p_{\eta}$ doesn't make sense (to me). Are you taking these figures directly from your course materials?

The figure showing the equation for $p_\eta (y)$ has "$x = f_1^{-1}(x)$" instead of "$x = f_1^{-1}(y)$", so it has errors also.

I think the simplest way to understand why the "transformation" approach works is to first study the "distribution method", which finds the cumulative distribution. Have you studied that method?

[PLAIN]http://pokit.org/get/efb0245c8d7ba024e4a8c3433dec0cb1.jpg

[PLAIN]http://pokit.org/get/90c6dc7a629e55a12b0e10c2e197f0f9.jpg

Here is how he got that.
The fact that the y goes up to 1 is ok with me. But the, f(y) is not.

 

[Bassalisk]

actually it does make sense. Because this is made out of 2 monotonous function, and the problem is ambiguity. The first part [0,pi/2] it goes from ~0.3 up until one and goes up to infinity. The second time it goes back from infinity to ~0.3. Add that up and will get ~0.6. I understand everything now. Thank you for effort anyway. You pointed out what I should be looking for. I did that in matlab, and everything adds up.

 

[Bassalisk]

Nope false alarm, I am clueless again. Matlab is showing the stuff we are seeing, when I am transforming these variables but I don't get it :D

 

[bpet]

Nope false alarm, I am clueless again. Matlab is showing the stuff we are seeing, when I am transforming these variables but I don't get it :D

It's much easier if you look at the CDF: F(x) = P[X<=x] (which Stephen hinted at in post #2).

HTH