Discussion
Let \(X\) be a positive continuous random variable. Consider the transformation \(Y = X^{4}\). Then, the Jacobian of the inverse transformation is
Answer: C
The Jacobian is \(\frac{1}{4}\, y^{-\frac{3}{4}}\).
Since \(X \gt 0\), the inverse transformation is \(x = y^{\frac{1}{4}}\), and its Jacobian is the derivative of \(x\) with respect to \(y\):
\[ \begin{aligned} J = \frac{dx}{dy} &= \frac{1}{4}\, y^{\frac{1}{4} - 1} \\ &= \frac{1}{4}\, y^{-\frac{3}{4}} \end{aligned} \]
This is the factor used in \(f_{Y}(y) = f_{X}(y^{1/4}) \cdot \frac{1}{4} y^{-3/4}\). The option \(y^{\frac{1}{4}}\) is the inverse transformation itself, not its derivative. \(\frac{3}{4} y^{-\frac{3}{4}}\) uses the wrong constant, and \(y^{\frac{3}{4}}\) has the wrong sign in the exponent.
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