Question 1179729
Here's how to find the method of moments estimators for α and β in a beta distribution:

**1. Recall the Moments of the Beta Distribution:**

For a beta distribution with parameters α and β, the first two moments are:

* E[X] = α / (α + β)
* E[X²] = (α(α + 1)) / ((α + β)(α + β + 1))

**2. Calculate the Sample Moments:**

We need to calculate the first two sample moments:

* First sample moment (X̄): X̄ = (1/n) * Σ[i=1 to n] Xi
* Second sample moment (M2): M2 = (1/n) * Σ[i=1 to n] Xi²

**3. Set Up the System of Equations:**

Equate the population moments to the sample moments:

* α / (α + β) = X̄
* (α(α + 1)) / ((α + β)(α + β + 1)) = M2

**4. Solve for α and β:**

Let's solve this system of equations:

* From the first equation, we have: α = X̄(α + β) => α(1 - X̄) = X̄β => β = α(1 - X̄) / X̄.
* Substitute this expression for β into the second equation:

    (α(α + 1)) / ((α + α(1 - X̄) / X̄)(α + α(1 - X̄) / X̄ + 1)) = M2

    Simplify:

    (α(α + 1)) / ((α/X̄)(α/X̄ + 1)) = M2

    (α(α + 1)X̄²) / (α(α + X̄)) = M2

    (α + 1)X̄² / (α + X̄) = M2

    (α + 1)X̄² = M2(α + X̄)

    αX̄² + X̄² = αM2 + M2X̄

    α(X̄² - M2) = M2X̄ - X̄²

    α = (M2X̄ - X̄²) / (X̄² - M2)

    α = X̄²(M2/X̄ - 1) / X̄²(1 - M2/X̄²)

    α = (M2/X̄ - 1) / (1 - M2/X̄²)

    α = (X̄M2 - X̄²) / (X̄² - M2)

    α = X̄(M2 - X̄) / (X̄² - M2)

* Now, substitute the expression for α back into the equation for β:

    β = α(1 - X̄) / X̄

    β = [(M2X̄ - X̄²) / (X̄² - M2)] * (1 - X̄) / X̄

    β = (M2 - X̄)(1 - X̄) / (X̄² - M2)

**5. Express in Terms of X̄ and S² (Sample Variance):**

We can express M2 in terms of the sample mean (X̄) and sample variance (S²):

* S² = M2 - X̄²
* M2 = S² + X̄²

Substitute this into the expressions for α and β:

* α = X̄(S² + X̄² - X̄) / (X̄² - (S² + X̄²)) = X̄(S² + X̄(X̄-1)) / (-S²)

    α = X̄(1-X̄)/S² * X̄

* β = (S² + X̄² - X̄)(1 - X̄) / (-S²)

    β = (1-X̄)(1-(X̄(1-X̄)/S²))

* α = X̄(1 - X̄) / S² - X̄

* β = (1-X̄)(1- (X̄(1-X̄)/S²))

* α = X̄(1 - X̄) / S²

* β = (1 - X̄)(1 - (X̄(1 - X̄) / S²))

* β = (1-X̄)α/X̄ = α(1-X̄)/X̄

**Final Answer:**

The method of moments estimators for α and β are:

* α̂ = X̄(1 - X̄) / S²
* β̂ = (1 - X̄) / X̄ * α̂ = (1 - X̄) / X̄ * [X̄(1 - X̄) / S²] = (1 - X̄)² / S²

Where:

* X̄ is the sample mean.
* S² is the sample variance.