How to transform a probability density function?

In summary, the conversation discusses a transformation of a probability density function from one interval to another while maintaining the function as a probability density function. The transformation involves a simple linear transformation and a substitution, which preserves the integral. The resulting function is similar to another function with different notation, but they coincide due to the identity sin2(x) = 1/2*(1 - cos(2x)).
  • #1
Ad VanderVen
169
13
TL;DR Summary
Transforming a trimodal probability density function with support [0;3*Pi] to a trimodal probability density function with support [(3/2)*Pi;(15/2)*Pi] .
I have the following probability density function (in Maple notation):

f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi]

Now I want to transform x so that

0 -> (3/2) * Pi
and
3 * Pi -> (15/2) * Pi

and the new function is still a probability density function.

How should I do that transformation?
 
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  • #2
Hi,
Ad VanderVen said:
0 -> (3/2) * Pi
and
3 * Pi -> (15/2) * Pi
It is a simple linear transformation.
So if we define ##x_0=0,\ x_1 = {3\over 2}\pi,\quad x_0'= 3\pi,\ x_1'={15\over2}\pi \;,\ ## then: $$x'= {x-x_0\over x_1-x_0} (x_1'-x'_0) \ \ {\sf and} \ \ dx'= {x'_1-x'_0\over x_1-x_0}\ dx$$

So you substitute ##\displaystyle {x = {x' - x_0'\over x_1'-x_0'}(x_1-x_0)}## and

with a simple factor ##{dx/ dx'}## you preserve the integral.

1591797960062.png
 
  • Like
Likes Dale
  • #3
Thanks a lot for your clear answer.

I have performed the transformation and I got (in Maple notation):

g (x) = (1/2) * (1 / ((3/2) * Pi)) * (sin ((1/2) * (x- (3/2) * Pi))) ** 2

with support [(3/2) * Pi; (15/2) * Pi)]

but this function coincides completely with the following function:

h (x) = (1 / (6 * Pi)) * (sin (x) +1)

also with support [(3/2) * Pi; (15/2) * Pi)]

Now g (x) is a quadratic function and h (x) is not. How can they coincide?
 
  • #4
sin2(x) = 1/2*(1 - cos(2x))
 
  • #5
Thank you very much. :bow:
 

1. What is a probability density function (PDF)?

A probability density function is a mathematical function that describes the likelihood of a random variable taking on a certain value. It is used to model continuous random variables and can be graphed as a curve on a coordinate plane.

2. Why would someone want to transform a PDF?

PDF transformation is often used in statistics and data analysis to convert a PDF into a more useful form for analysis. This can help to simplify the data or make it easier to interpret.

3. How do you transform a PDF?

The process of transforming a PDF involves applying a mathematical function to the original PDF. This can include taking the logarithm, square root, or other transformations. The resulting transformed PDF will have a different shape and may be more suitable for analysis.

4. What are some common transformations used for PDFs?

Some common transformations used for PDFs include the logarithmic, exponential, and power transformations. These can help to normalize the data or make it easier to interpret. Other transformations may be used depending on the specific data and analysis being performed.

5. What are some potential limitations of transforming a PDF?

While transforming a PDF can be useful, it is important to keep in mind that it may alter the original data and may not accurately represent the true distribution. It is also important to choose the appropriate transformation carefully, as certain transformations may not be suitable for all types of data.

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