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Let X1 and X2 denote a random sample of size 2 from a distribution with pdf f(x) = 1/2, 0 < x < 1, zero elsewhere. Find the distribution function and the pdf of Y = X1/X2.© BrainMass Inc. brainmass.com March 6, 2023, 1:13 pm ad1c9bdddf
Y = X1/X2
Let Z = X1
=> X1 = Z and X2 = X1/Y = Z/Y
| dY/dX1 dY/dX2|
=> J(X1,X2) = dY/dX1 * dZ/dX2 - dY/dX2 * dZ/dX1
dY/dX1 = 1/X2; dY/X2 = -X1/X2^2; dZ/dX1 = 1; dZ/dX2 = 0
=> J(X1, X2) = 0 - (-X1/X2^2) * 1 = X1/X2^2
=> |J(X1,X2)| = X1/X2^2
I'm assuming Xa and X2 ...
The solution answers the question(s) below. The sample distribution functions are examined.