Question

Suppose X and Y are independent random variables, where $X \sim Exp(\frac{1}{\theta})$ and $Y \sim Uniform(0, \theta)$. Let $\hat{\theta} = aX + bY$ denote a class of estimators of $\theta$, where a and b are constants. a) Determine the bias of $\hat{\theta}$. b) Determine the variance of $\hat{\theta}$. c) Determine the mean squared error of $\hat{\theta}$. d) Determine the condition involving a and b such that $\hat{\theta}$ is unbiased for $\theta$. e) Determine the values of a and b such that $\hat{\theta}$ is unbiased for $\theta$ with the smallest variance. f) Determine the mean, variance and MSE for $\hat{\theta}$ when a = 2 and b = 3.

          Suppose X and Y are independent random variables, where $X \sim Exp(\frac{1}{\theta})$ and $Y \sim Uniform(0, \theta)$. Let $\hat{\theta} = aX + bY$ denote a class of estimators of $\theta$, where a and b are constants.
a) Determine the bias of $\hat{\theta}$.
b) Determine the variance of $\hat{\theta}$.
c) Determine the mean squared error of $\hat{\theta}$.
d) Determine the condition involving a and b such that $\hat{\theta}$ is unbiased for $\theta$.
e) Determine the values of a and b such that $\hat{\theta}$ is unbiased for $\theta$ with the smallest variance.
f) Determine the mean, variance and MSE for $\hat{\theta}$ when a = 2 and b = 3.

Suppose X and Y are independent random variables, where X ∼ Exp((1)/(θ)) and Y ∼ Uniform(0, θ). Let θ̂ = aX + bY denote a class of estimators of θ, where a and b are constants.
a) Determine the bias of θ̂.
b) Determine the variance of θ̂.
c) Determine the mean squared error of θ̂.
d) Determine the condition involving a and b such that θ̂ is unbiased for θ.
e) Determine the values of a and b such that θ̂ is unbiased for θ with the smallest variance.
f) Determine the mean, variance and MSE for θ̂ when a = 2 and b = 3.

Added by Kenneth C.

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