Finding the pdf of a transformed univariate random variable

  • #1
Hamiltonian
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TL;DR Summary
Confused as to how to obtain the cdf of a transformed random variable.
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The above theorem is trying to find the pdf of a transformed random variable, it attempts to do so by "first principles", starting by using the definition of cdf, I don't understand why they have a in the integral wouldn't be the correct integral for the cdf of Y.
 
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  • #2
No. The way gets into the equation is only in the limit of the integral, . Suppose . What is the probability (density) of that? It is the probability (density) of that associated value, which is . So those are the probabilities that we want to total for the integral. (Notice that the integral is with respect to , not )

PS. I may have ignored the situation where multiple values give the same . That still works out because the integral limit allows the associated densities to be summed
 
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  • #3
Suppose and we know the pdf of X. To get the pdf of Y, we can find the CDF of Y then differentiate it wrt to y to get the pdf.

And depending on X you have to do an appropriate manipulation to get the cdf of Y. Here's and example, deriving the pdf of a function with a deg of freedom of one. This variable here is just , Z is a standard normal distribution. The pdf of Z is a bit complex but you can find it here.
https://www.thoughtco.com/normal-distribution-bell-curve-formula-3126278 lets call this function f(x).
And cdf of f is and called F(x). Let X be a standard normal variable, and . So

which is differentiating wrt y
 
  • #4
Apologies the website gave gives the formula for the general normal distribution for the standard normal, take in the equation
 

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