Question

Let X be a discrete random variable whose PMF is $\qquad p(0) = \frac{2\theta}{3}, \quad p(1) = \frac{\theta}{3}, \quad p(2) = \frac{2(1-\theta)}{3}, \quad p(3) = \frac{1-\theta}{3}$. Suppose that $X_1, \dots, X_n$ are iid samples of X ($n > 0)$. (a) Find the MME for $\theta$. [10 marks] (b) Is the MME unbiased? Is it asymptotically unbiased? [10 marks] (c) Find the MLE for $\theta$. [10 marks] (d) Is the MLE unbiased? Is it asymptotically unbiased? [10 marks]

          Let X be a discrete random variable whose PMF is
$\qquad p(0) = \frac{2\theta}{3}, \quad p(1) = \frac{\theta}{3}, \quad p(2) = \frac{2(1-\theta)}{3}, \quad p(3) = \frac{1-\theta}{3}$.
Suppose that $X_1, \dots, X_n$ are iid samples of X ($n > 0)$.
(a) Find the MME for $\theta$. [10 marks]
(b) Is the MME unbiased? Is it asymptotically unbiased? [10 marks]
(c) Find the MLE for $\theta$. [10 marks]
(d) Is the MLE unbiased? Is it asymptotically unbiased? [10 marks]

Let X be a discrete random variable whose PMF is
p(0) = (2θ)/(3), p(1) = (θ)/(3), p(2) = (2(1-θ))/(3), p(3) = (1-θ)/(3).
Suppose that X1, …, Xn are iid samples of X (n > 0).
(a) Find the MME for θ. [10 marks]
(b) Is the MME unbiased? Is it asymptotically unbiased? [10 marks]
(c) Find the MLE for θ. [10 marks]
(d) Is the MLE unbiased? Is it asymptotically unbiased? [10 marks]

Added by Alexis M.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Areen Dabadghav

Step 1

To find the MME, we need to equate the sample moments to the population moments. The first population moment is E(X) = Σx * p(x), where x is the value of the random variable and p(x) is the probability mass function. In this case, we have: E(X) = 0 * p(0) + 1 * Show more…

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I'll give you Thumbs up if your answer is correct and clearly written! Let X be a discrete random variable whose PMF is 20 p(0) = 3 0 p(1) = 3 2(1 0) p(2) = 3 1 _ 0 2(3) = 3 Suppose that Xi, Xn are iid samples of X (n > 0). Find the MME for 0. [10 marks] Is the MME unbiased? Is it asymptotically unbiased? [10 marks] Find the MLE for 0. [10 marks] Is the MLE unbiased? Is it asymptotically unbiased? [10 marks]