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I tried to substitue variable x of one equation with another variable z (z=nx/(nx+m)). See as follow:

Solve[z == n*x/(n*x + m), x]
x -> -((m z)/(n (-1 + z)))

Then:

PDF[FRatioDistribution[n, m], x] /. x -> -((m z)/(n (-1 + z)))
 (m^(m/2) n^(n/2) (-((m z)/(n (-1 + z))))^(-1 + n/2) (m - (m z)/(-1 + z))^(1/2 
  (-m - n)))/Beta[n/2, m/2]

How can I collect the term z and 1-z together? Factually, if collecting the results by `z and1-z`, it can be very simple (that is Beta distribution). Any suggestion will be greatly appreciated.

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    $\begingroup$ Just FullSimplify[..., Assumptions -> {n > 0, m > 0}] $\endgroup$ – ybeltukov Oct 13 '13 at 22:05
  • $\begingroup$ Thaks your reply. it helps. $\endgroup$ – user2069518 Oct 14 '13 at 5:13
  • $\begingroup$ @ybeltukov answer? $\endgroup$ – Yves Klett Oct 14 '13 at 17:45
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FullSimplify with Assumptions makes what you want

FullSimplify[PDF[FRatioDistribution[n, m], x] /. x -> -((m z)/(n (-1 + z))), 
  Assumptions -> {n > 0, m > 0}]
(n (1 - z)^(1 + m/2) z^(-1 + n/2))/(m Beta[n/2, m/2])
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