Question:
Let \( X \) be a continuous random variable with the probability density function
\[
f(x) = \frac{1}{3} x^2 e^{-x^2}, \quad x>0
\]
Then the distribution of the random variable
\[
W = 2X^2 \quad \text{is}
\]
Let \( X \) be a continuous random variable with the probability density function
\[
f(x) = \frac{1}{3} x^2 e^{-x^2}, \quad x>0
\]
Then the distribution of the random variable
\[
W = 2X^2 \quad \text{is}
\]
Show Hint
For transformations of random variables, use the properties of the chi-square distribution to determine the distribution of the new variable.
Updated On: Dec 12, 2025
- \( \chi^2_2 \)
- \( \chi^4_4 \)
- \( \chi^2_4 \)
- \( \chi^2_8 \)
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The Correct Option is D
Solution and Explanation
Step 1: Transform the variable.
The random variable \( W = 2X^2 \) is a scaled version of \( X^2 \). Since \( X^2 \) follows a chi-square distribution with 1 degree of freedom, \( W = 2X^2 \) follows a chi-square distribution with 2 degrees of freedom.
Step 2: Conclusion.
The correct answer is (A) \( \chi^2_2 \).
The random variable \( W = 2X^2 \) is a scaled version of \( X^2 \). Since \( X^2 \) follows a chi-square distribution with 1 degree of freedom, \( W = 2X^2 \) follows a chi-square distribution with 2 degrees of freedom.
Step 2: Conclusion.
The correct answer is (A) \( \chi^2_2 \).
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