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: Write the given transformation.
The transformation is
\[ Y=X^4 \]
Since \(X\) is positive, the inverse transformation is unique.

Step 2: Find the inverse transformation.
From
\[ Y=X^4 \] we get
\[ X=Y^{\frac{1}{4}} \]
Thus, the inverse transformation is
\[ x=y^{\frac{1}{4}} \]

Step 3: Differentiate the inverse transformation.
The Jacobian of the inverse transformation is
\[ \left|\frac{dx}{dy}\right| \]
Now,
\[ x=y^{\frac{1}{4}} \]
Differentiate with respect to \(y\):
\[ \frac{dx}{dy} = \frac{1}{4}y^{\frac{1}{4}-1} \]
\[ \frac{dx}{dy} = \frac{1}{4}y^{-\frac{3}{4}} \]

Step 4: Identify the Jacobian.
Since \(y>0\), the derivative is positive.
Therefore,
\[ \left|\frac{dx}{dy}\right| = \frac{1}{4}y^{-\frac{3}{4}} \]

Step 5: Match with the options.
The expression
\[ \frac{1}{4}y^{-\frac{3}{4}} \] matches option (C).

Step 6: Final conclusion.
Hence, the Jacobian of the inverse transformation is
\[ \boxed{\frac{1}{4}y^{-\frac{3}{4}}} \]