Question 1179730
Here's how to find the moment estimator for α:

**1. Find the First Population Moment (E[X]):**

The first population moment is the expected value (mean) of the distribution:

E[X] = ∫[-1, 1] x * f(x, α) dx
E[X] = ∫[-1, 1] x * (1 + αx / 2) dx
E[X] = ∫[-1, 1] (x + αx² / 2) dx
E[X] = [x²/2 + αx³/6] evaluated from -1 to 1
E[X] = [(1/2 + α/6) - (1/2 - α/6)]
E[X] = α/3

**2. Find the First Sample Moment (X̄):**

The first sample moment is the sample mean, denoted as X̄:

X̄ = (1/n) * Σ[i=1 to n] Xi

**3. Equate Population and Sample Moments:**

Set the first population moment equal to the first sample moment:

E[X] = X̄
α/3 = X̄

**4. Solve for α:**

α = 3 * X̄

**Therefore, the moment estimator for α is:**

α̂ = 3 * X̄ = (3/n) * Σ[i=1 to n] Xi