Question

Let \(X\) be a positive continuous random variable. Consider the transformation \(Y=X^4\). Then, the Jacobian of the inverse transformation is

• A. \(\frac{3}{4}y^{-\frac{3}{4}}\)
• B. \(\frac{3}{y^4}\)
• C. \(\frac{1}{4}y^{-\frac{3}{4}}\)
• D. \(\frac{1}{y^4}\)

Hint:

For transformation questions, first find the inverse function and then compute the absolute value of its derivative.

Answer 1

The Correct Option is C

Step 1: Invert the transformation.
We have $Y=X^4$ with $X>0$, so $X=Y^{1/4}$, that is $x=y^{1/4}$.

Step 2: Recall what the Jacobian is.
For a one variable change, the Jacobian of the inverse is $\left|\dfrac{dx}{dy}\right|$.

Step 3: Differentiate.
From $x=y^{1/4}$, \[ \frac{dx}{dy}=\frac14 y^{-3/4}. \]

Step 4: Drop the sign.
Since $y>0$ this is already positive, so the Jacobian is $\dfrac14 y^{-3/4}$.

Step 5: Match.
This is option (C).
\[ \boxed{\frac14 y^{-3/4}} \]