Question

(40%) Let $g(x)$ be an unbiased estimator for the estimation of $\theta$ with the observations of the random variables $X_1, X_2, \dots, X_N$ under the joint PDF \\ $f(X_1 = x_1, X_2 = x_2, \dots, X_N = x_N; \theta)$ denoted by $f(x; \theta)$. \\ (a). (10%) Using the unbiased property, show that $\int_{-\infty}^{\infty} (g(x) - \theta) \frac{\partial f(x; \theta)}{\partial \theta} dx = 1$. \\ Hint: Unbiased property: $E\{g(x) - \theta\} = \int_{-\infty}^{\infty} (g(x) - \theta) f(x; \theta) dx = 0$.

          (40%) Let $g(x)$ be an unbiased estimator for the estimation of $\theta$ with the observations of the random variables $X_1, X_2, \dots, X_N$ under the joint PDF \\
$f(X_1 = x_1, X_2 = x_2, \dots, X_N = x_N; \theta)$ denoted by $f(x; \theta)$. \\
(a). (10%) Using the unbiased property, show that $\int_{-\infty}^{\infty} (g(x) - \theta) \frac{\partial f(x; \theta)}{\partial \theta} dx = 1$. \\
Hint: Unbiased property: $E\{g(x) - \theta\} = \int_{-\infty}^{\infty} (g(x) - \theta) f(x; \theta) dx = 0$.

$(40%) Let $g(x)$ be an unbiased estimator for the estimation of $\theta$ with the observations of the random variables $X1, X2, \dots, XN$ under the joint PDF \\ f(X1 = x1, X2 = x2, …, XN = xN; θ) denoted by f(x; θ). (a). (10%) Using the unbiased property, show that $\int{-\infty}^{\infty} (g(x) - \theta) \frac{\partial f(x; \theta)}{\partial \theta} dx = 1$. \\ Hint: Unbiased property: E{g(x) - θ} = ∫-∞^∞ (g(x) - θ) f(x; θ) dx = 0.$

Added by Bryan J.

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