Let \( x_1 = -2, x_2 = 1 \) and \( x_3 = -1 \) be the observed values of a random sample of size three from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \frac{1}{2 \theta + 1}, & x \in \{-\theta, -\theta + 1, \dots, 0, \dots, \theta \}, \\ 0, & \text{otherwise}, \end{cases} \] where \( \theta \in \{ 1, 2, \dots \} \) is the unknown parameter. Then the method of moment estimate of \( \theta \) is
Let \( x_1 = -2, x_2 = 1 \) and \( x_3 = -1 \) be the observed values of a random sample of size three from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \frac{1}{2 \theta + 1}, & x \in \{-\theta, -\theta + 1, \dots, 0, \dots, \theta \}, \\ 0, & \text{otherwise}, \end{cases} \] where \( \theta \in \{ 1, 2, \dots \} \) is the unknown parameter. Then the method of moment estimate of \( \theta \) is
Show Hint
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- 3
- 4
The Correct Option is B
Solution and Explanation
Step 1: Use the method of moments.
The method of moments involves equating the sample moments to the theoretical moments. The first moment (mean) of the distribution is:
\[
E(X) = \frac{1}{2\theta + 1} \sum_{x=-\theta}^{\theta} x = 0,
\]
which is true by symmetry.
Step 2: Use the second moment.
The second moment (variance) of the distribution is:
\[
\text{Var}(X) = E(X^2) - (E(X))^2 = \frac{1}{2\theta + 1} \sum_{x=-\theta}^{\theta} x^2.
\]
Using the sample variance, compute the method of moments estimate for \( \theta \).
Step 3: Conclusion.
The method of moment estimate for \( \theta \) is (B) 2.
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