probability density function of standard Cauchy variate and cauchy variate having two parameters

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  1. Study Nutz
  2. probability density function of standard Cauchy variate
  3. probability density function of cauchy variate having two parameters
  4. relation between standard cauchy variate and cauchy variate with two parameters
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Автор

Hi! Is there a lambda missing in the numerator of Gy(y)? Should it be lambda^2?

pablojvazquez
Автор

Let me expose to you what I get from your video:

1) You begin with a standard Cauchy distribution, with a pdf >> f(x)=1/[π·(1+x^2)]
2) You then substitute x by (y-μ)/ɣ. Ok, it's a kind of normalization of y. And, as long as μ and ɣ are constants, and y is derived from both these constants and x, then f(x)=f((y-μ)/ɣ)
3) So, we can write f(x)=1/[π·(1+(y-μ)^2/ɣ^2)]
4) The relationship between x and y will be given by dx=1/ɣ*dy
5) The pdf is the derivative of cdf, so dF(x)/dx=f(x) and dF(x)=f(x)*dx
6) Substituting dx by 1/ɣdy we get dF(x)=f(x)*1/ɣdy
7) Then dF(x)/dy = f(x)1/ɣ = 1/[π·(1+(y-μ)^2/ɣ^2)]*1/ɣ = 1/π[ɣ/(ɣ^2+(y-μ)^2)]
8) dF(x)/dy = f(y) = (1/π)·[ɣ/(ɣ^2+(y-μ)^2)], right?

But why dF(x)/dy = f(y)?

pablojvazquez