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Inferential Statisticshard
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A researcher investigates a model where the observed data follows a distribution with density f(x∣θ)=θ2xe−θxf(x|\theta) = \theta^2 x e^{-\theta x}f(x∣θ)=θ2xe−θx for x>0x > 0x>0. If the researcher derives the log-likelihood ℓ(θ)=2nln⁡θ+∑ln⁡xi−θ∑xi\ell(\theta) = 2n \ln \theta + \sum \ln x_i - \theta \sum x_iℓ(θ)=2nlnθ+∑lnxi​−θ∑xi​, what is the Fisher Information I(θ)I(\theta)I(θ) for a single observation?