Question

Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?

• A. \( \xi_{0.75} \geq 5 \)
• B. \( \xi_{0.75} \leq 4 \)
• C. \( \xi_{0.25} \geq 4 \)
• D. \( \xi_{0.25} = 2 \)

Answer 1

The Correct Option is B

1. Quantiles: - The \( p \)-th quantile \( \xi_p \) satisfies: \[ P(Y \leq \xi_p) = p. \] - Given \( P(Y > 0) = 1 \), all quantiles are positive. 
2. Expected Value Constraint: - \( E(Y) = 1 \) implies the distribution is concentrated around small values of \( Y \). 
Thus, \( \xi_{0.75} \) is likely to be less than or equal to 4. 
3. Analyze the Statements: 
- (A): \( \xi_{0.75} \geq 5 \): Not true as \( E(Y) = 1 \), so higher quantiles are unlikely to exceed 4 significantly. 
- (B): \( \xi_{0.75} \leq 4 \): Correct. 
- (C): \( \xi_{0.25} \geq 4 \): Unlikely as the lower quantiles are closer to 0.
- (D): \( \xi_{0.25} = 2 \): This is not guaranteed.