Question 1179730: Let X1,...,Xn be a random sample from a distribution with pdf
f(x, α) = 1 + αx / 2 , −1 ≤ x ≤ 1, and − 1 ≤ α ≤ 1.
Find the moment estimators for α.
Thank you...
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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
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