How to Determine the Transforming Function g(.) from PDFs and CDFs?

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SUMMARY

To determine the transforming function g(.) from the Probability Density Functions (PDFs) of two random variables X and Y, one must utilize the Cumulative Distribution Functions (CDFs) F_X(x) and F_Y(y). The process is simplified by assuming that g(.) is a one-to-one and monotonic function. This approach allows for a clearer specification of g(.) based on the relationship between the CDFs of the random variables.

PREREQUISITES
  • Understanding of Probability Density Functions (PDFs)
  • Knowledge of Cumulative Distribution Functions (CDFs)
  • Familiarity with monotonic functions
  • Basic concepts of random variables
NEXT STEPS
  • Study the properties of one-to-one and monotonic functions in probability theory
  • Learn how to derive CDFs from PDFs for random variables
  • Explore transformations of random variables in statistical analysis
  • Investigate examples of transforming functions in practical applications
USEFUL FOR

Statisticians, data analysts, and students studying probability theory who need to understand the relationship between random variables through transformation functions.

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Hello,

Suppose that a random variable Y is formed by transforming another random variable X by using the tranforming function g(.). That is:

Y=\,g(X)

Now, given that we have the Probabililty Density Function (PDF) of both RVs: f_Y(y)\mbox{ and }f_X(x), how can we specify g(.)? I didn't give an exact example because I just need to know the procedure.

Thanks in advance
 
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S_David said:
Hello,

Suppose that a random variable Y is formed by transforming another random variable X by using the tranforming function g(.). That is:

Y=\,g(X)

Now, given that we have the Probabililty Density Function (PDF) of both RVs: f_Y(y)\mbox{ and }f_X(x), how can we specify g(.)? I didn't give an exact example because I just need to know the procedure.

Thanks in advance

It'll be a bit easier to use the CDFs and assuming g is one-to-one & monotonic:

F_X(x) = Prob(X<=x) etc.
 
bpet said:
It'll be a bit easier to use the CDFs and assuming g is one-to-one & monotonic:

F_X(x) = Prob(X<=x) etc.

Ok, then?
 

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