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3. Let \(X_1, X_2, \dots, X_n\) be a random sample from a distribution with the probability density function: \(f(x) = \begin{cases} \frac{e^{-x/\theta}}{\theta} & \text{for } x > 0\\ 0 & \text{otherwise} \end{cases}\) where \(\theta > 0\) is a parameter. (a) Derive the maximum likelihood estimator, \(\hat{\theta}\), of \(\theta\). (6 marks) (b) Check whether \(\hat{\theta}\) is an unbiased estimator of \(\theta\). (4 marks) (c) Calculate the variance of \(\hat{\theta}\). (7 marks) (d) Explain whether \(\hat{\theta}\) is a consistent estimator of \(\theta\). (3 marks) Hint: You may find the following integral useful: \(\int_0^\infty x^n e^{-x/\theta} dx = n! \theta^{n+1}\)

          3. Let \(X_1, X_2, \dots, X_n\) be a random sample from a distribution with the probability
density function:

\(f(x) = \begin{cases} \frac{e^{-x/\theta}}{\theta} & \text{for } x > 0\\ 0 & \text{otherwise} \end{cases}\)

where \(\theta > 0\) is a parameter.

(a) Derive the maximum likelihood estimator, \(\hat{\theta}\), of \(\theta\).
(6 marks)
(b) Check whether \(\hat{\theta}\) is an unbiased estimator of \(\theta\).
(4 marks)
(c) Calculate the variance of \(\hat{\theta}\).
(7 marks)
(d) Explain whether \(\hat{\theta}\) is a consistent estimator of \(\theta\).
(3 marks)
Hint: You may find the following integral useful:

\(\int_0^\infty x^n e^{-x/\theta} dx = n! \theta^{n+1}\)

3. Let X1, X2, …, Xn be a random sample from a distribution with the probability
density function:

f(x) = (e^-x/θ)/(θ) for x > 0
0 otherwise

where θ > 0 is a parameter.

(a) Derive the maximum likelihood estimator, θ̂, of θ.
(6 marks)
(b) Check whether θ̂ is an unbiased estimator of θ.
(4 marks)
(c) Calculate the variance of θ̂.
(7 marks)
(d) Explain whether θ̂ is a consistent estimator of θ.
(3 marks)
Hint: You may find the following integral useful:

∫0^∞ x^n e^-x/θ dx = n! θ^n+1

Added by Joe R.

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