Transformation of random variable

In summary: I found here:In summary, the problem is asking for a graph of a probability density function, which is shown in the final graph.
  • #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?
 
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  • #2
Bassalisk said:
This is pη 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.
 
  • #3
Stephen Tashi said:
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
 
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  • #4
The last two graphs make sense. The graph of [itex] p_{\eta}[/itex] doesn't make sense (to me). Are you taking these figures directly from your course materials?

The figure showing the equation for [itex] p_\eta (y) [/itex] has "[itex] x = f_1^{-1}(x)[/itex]" instead of "[itex]x = f_1^{-1}(y)[/itex]", 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?
 
  • #5
Stephen Tashi said:
The last two graphs make sense. The graph of [itex] p_{\eta}[/itex] doesn't make sense (to me). Are you taking these figures directly from your course materials?

The figure showing the equation for [itex] p_\eta (y) [/itex] has "[itex] x = f_1^{-1}(x)[/itex]" instead of "[itex]x = f_1^{-1}(y)[/itex]", 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.
 
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  • #6
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.
 
  • #7
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
 
  • #8
Bassalisk said:
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
 

1. What is the definition of transformation of random variable?

The transformation of random variable refers to the process of changing or manipulating the values of a random variable to create a new random variable. This can be done through mathematical operations such as addition, multiplication, or applying functions to the original random variable.

2. Why is transformation of random variable important in statistics?

The transformation of random variable is important in statistics because it allows for the simplification and improvement of data analysis. By transforming a random variable, we can often make the data more normally distributed, which is a key assumption for many statistical tests. It also allows us to better understand the relationship between variables and make more accurate predictions.

3. What are some common transformations used in statistics?

Some common transformations used in statistics include logarithmic, exponential, square root, and inverse transformations. These transformations are often used to deal with skewed data, where the original variable does not follow a normal distribution.

4. How do you determine which transformation to use?

The choice of transformation depends on the distribution of the original data and the goal of the analysis. For example, if the data is positively skewed, a logarithmic transformation may be appropriate to make it more symmetrical. It is important to also consider the interpretation of the transformed data and choose a transformation that makes the most sense in the context of the research question.

5. Are there any risks associated with transformation of random variable?

Yes, there are some risks associated with transformation of random variable. It can potentially introduce bias or distort the relationship between variables. It is important to carefully consider the effects of transformation on the data and to choose an appropriate transformation method to minimize these risks.

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