Question

Let \( X \) and \( Y \) be random variables having chi-square distributions with 6 and 3 degrees of freedom respectively. Then, which of the following statements is TRUE?

• A. \( P(X>0.7)>P(Y>0.7) \)
• B. \( P(X>0.7)<P(Y>0.7) \)
• C. \( P(X>3)<P(Y>3) \)
• D. \( P(X<6)>P(Y<6) \)

Hint:

For chi-square distributions, increasing degrees of freedom shifts the curve rightward; smaller df gives higher probability near zero.

Answer 1

The Correct Option is A

Step 1: Recall properties of chi-square distribution.
For a chi-square variable with \( k \) degrees of freedom, the mean is \( k \) and variance is \( 2k \). As \( k \) increases, the distribution becomes more symmetric and spreads to the right.
Step 2: Compare \( X \sim \chi^2(6) \) and \( Y \sim \chi^2(3) \).
- \( X \) has a larger mean (6) than \( Y \) (3). - For the same value of \( x \), the probability \( P(X<x) \) will be greater when \( x \) is close to \( Y \)'s mean because \( X \)'s curve is shifted right.
Step 3: Check each option.
(A) \( P(X>0.7)>P(Y>0.7) \): False, since \( Y \) has a lower mean, its right tail probability is larger for small \( x \). (B) \( P(X>0.7)<P(Y>0.7) \): True but not the most precise comparison. (C) \( P(X>3)<P(Y>3) \): False, at \( x = 3 \), \( X \)'s mean is larger, so probability of exceeding 3 is higher for \( X \). (D) \( P(X<6)>P(Y<6) \): True, since 6 is near the mean of \( X \), \( P(X<6) \approx 0.5 \), while for \( Y \), 6 is far right tail, so \( P(Y<6)<0.5 \).
Step 4: Conclusion.
Thus, \( P(X<6)>P(Y<6) \) is correct. Final Answer: \[ \boxed{P(X<6)>P(Y<6)} \]