SOLUTION: Let ε (for “error”) be a random variable with E(ε) = 0, and Var(ε) = σ ^2. Define the random variable, X = μ + ε, where μ is a constant. Find E(X), Var(X), and E(ε^2).
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-> SOLUTION: Let ε (for “error”) be a random variable with E(ε) = 0, and Var(ε) = σ ^2. Define the random variable, X = μ + ε, where μ is a constant. Find E(X), Var(X), and E(ε^2).
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Question 1177297: Let ε (for “error”) be a random variable with E(ε) = 0, and Var(ε) = σ ^2. Define the random variable, X = μ + ε, where μ is a constant. Find E(X), Var(X), and E(ε^2).
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Given Information
E(ε) = 0
Var(ε) = σ²
X = μ + ε, where μ is a constant
1. Find E(X)
E(X) = E(μ + ε)
E(X) = E(μ) + E(ε) (Linearity of expectation)
E(X) = μ + 0 (Since E(μ) = μ and E(ε) = 0)
E(X) = μ
2. Find Var(X)
Var(X) = Var(μ + ε)
Var(X) = Var(ε) (Variance of a constant is 0, and Var(aX) = a²Var(X))
Var(X) = σ²
3. Find E(ε²)
We know that Var(ε) = E(ε²) - [E(ε)]².
Var(ε) = σ²
E(ε) = 0
Substitute these values:
σ² = E(ε²) - (0)²
σ² = E(ε²) - 0
σ² = E(ε²)
Therefore, E(ε²) = σ².
Answers
E(X) = μ
Var(X) = σ²
E(ε²) = σ²
