The fundamental assumptions of consumer preference theory include the properties of completeness, transitivity, reflexivity, non-satiation, continuity, and strict convexity. Let’s analyze each option with respect to these properties:
Option (C) violates the fundamental assumption of reflexivity, which states that any bundle x is always at least as good as itself (x ∼ x). Therefore, the statement “x is not indifferent to itself” is incorrect.
Let \( X_1, X_2 \) be a random sample from a population having probability density function
\[ f_{\theta}(x) = \begin{cases} e^{(x-\theta)} & \text{if } -\infty < x \leq \theta, \\ 0 & \text{otherwise}, \end{cases} \] where \( \theta \in \mathbb{R} \) is an unknown parameter. Consider testing \( H_0: \theta \geq 0 \) against \( H_1: \theta < 0 \) at level \( \alpha = 0.09 \). Let \( \beta(\theta) \) denote the power function of a uniformly most powerful test. Then \( \beta(\log_e 0.36) \) equals ________ (rounded off to two decimal places).
Let \( X_1, X_2 \) be a random sample from a distribution having probability density function