Let's break down the problem **step-by-step**: ## Problem Restatement Given a random sample \( (X_1, ..., X_n) \), determine which statistics converge in probability by the **Weak Law of Large Numbers (WLLN)** and the **Continuous Mapping Theorem (CMT)**, and for each, which moments must exist. --- ## **Step 1: Recall the Theorems** ### Weak Law of Large Numbers (WLLN) If \( X_1, ..., X_n \) are i.i.d. with mean \( \mu \) and finite variance, then: \[ \overline{X}_n = \frac{1}{n}\sum_{i=1}^n X_i \xrightarrow{P} \mu \] as \( n \to \infty \). ### Continuous Mapping Theorem (CMT) If \( Y_n \xrightarrow{P} \theta \), and \( g \) is continuous at \( \theta \), then \( g(Y_n) \xrightarrow{P} g(\theta) \). --- ## **Step 2: Analyze Each Statistic** ### (a) \( 12 - 1\overline{X} \) - \( \overline{X} \xrightarrow{P} E[X] \) by WLLN, if \( E[X] \) exists. - \( 12 - 1\overline{X} \) is a continuous function, so by CMT: \[ 12 - 1\overline{X} \xrightarrow{P} 12 - 1E[X] \] - **Moment needed**: \( E[X] \). --- ### (b) \( h(\overline{X}) \) - Same reasoning as above; by WLLN and CMT, \( h(\overline{X}) \xrightarrow{P} h(E[X]) \) if \( h \) is continuous at \( E[X] \). - **Moment needed**: \( E[X] \). --- ### (c) \( \max_i X_i \) - The maximum does **not** converge in probability to \( E[X] \) (unless \( X \) is degenerate). - The maximum converges to the **essential supremum** of the distribution (if it exists). - **WLLN does not apply**; this statistic does **not** converge by WLLN. --- ### (d) \( \frac{1}{n} \sum_{i=1}^n (27 - 1X_i^2 - 64 \cdot 1X_i^2) \) - This is a sample mean. By WLLN, it converges to the expected value if \( E[27 - 1X^2 - 64 \cdot 1X^2] \) exists. - **Moment needed**: \( E[X^2] \). --- ### (e) \( \frac{1}{n} \sum_{i=1}^n X_i^2 \) assuming \( E[X] > \) - Sample mean of \( X_i^2 \) converges to \( E[X^2] \) if \( E[X^2] \) exists. - **Moment needed**: \( E[X^2] \). --- ### (f) \( \frac{1}{n} \sum_{i=1}^n 1\{27 - 1X_i > \} \) - This is the sample proportion where \( 27 - 1X_i > \). - By WLLN, converges to \( P(27 - 1X > ) \). - **Moment needed**: None (just probability exists). --- ### (g) \( 121 - 1X_i Y_i \) - If \( (X_i, Y_i) \) are i.i.d., \( \frac{1}{n} \sum_{i=1}^n (121 - 1X_i Y_i) \) converges to \( 121 - E[X Y] \) if \( E[X Y] \) exists. - **Moment needed**: \( E[X Y] \). --- ## **Step 3: Summary Table** | Statistic | Converges by WLLN + CMT? | Moment Needed | |---------------------------------------------|--------------------------|----------------------| | (a) \( 12 - 1\overline{X} \) | Yes | \( E[X] \) | | (b) \( h(\overline{X}) \) | Yes (if \( h \) cont.) | \( E[X] \) | | (c) \( \max_i X_i \) | No | — | | (d) \( \frac{1}{n}\sum (27-1X_i^2-64\cdot 1X_i^2) \) | Yes | \( E[X^2] \) | | (e) \( \frac{1}{n} \sum X_i^2 \) | Yes | \( E[X^2] \) | | (f) \( \frac{1}{n} \sum 1\{27-1X_i > \} \) | Yes | None | | (g) \( 121 - 1X_iY_i \) | Yes | \( E[X Y] \) | --- ## **Final Answers** ### **Converge by WLLN + CMT:** - **(a), (b), (d), (e), (f), (g)** ### **Moments Needed:** - (a): \( E[X] \) - (b): \( E[X] \) - (d): \( E[X^2] \) - (e): \( E[X^2] \) - (f): None - (g): \( E[XY] \) **(c) does NOT converge by WLLN.**