Question

Let \( x_1 = 0, x_2 = 1, x_3 = 2, x_4 = 3 \) and \( x_5 = 0 \) be the observed values of a random sample of size 5 from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \frac{\theta}{3}, & x = 0, \\ \frac{2\theta}{3}, & x = 1, \\ \frac{1 - \theta}{2}, & x = 2, 3, \end{cases} \] where \( \theta \in [0,1] \) is the unknown parameter. Then the maximum likelihood estimate of \( \theta \) is 
 

• A. \( \frac{2}{5} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{5}{7} \)
• D. \( \frac{5}{9} \)

Hint:

When solving for the maximum likelihood estimate, first write the likelihood function, then differentiate with respect to \( \theta \), and solve for the value that maximizes the likelihood.

Answer 1

The Correct Option is B

Step 1: Write the likelihood function.
The likelihood function \( L(\theta) \) is the product of the probability mass functions for each observation: \[ L(\theta) = \prod_{i=1}^{5} f(x_i; \theta). \] Given the observations \( x_1 = 0, x_2 = 1, x_3 = 2, x_4 = 3, x_5 = 0 \), we substitute into the probability mass function: \[ L(\theta) = \left( \frac{\theta}{3} \right)^2 \cdot \left( \frac{2\theta}{3} \right) \cdot \left( \frac{1-\theta}{2} \right)^2. \] Simplifying: \[ L(\theta) = \frac{\theta^2 \cdot 2\theta \cdot (1-\theta)^2}{3^3 \cdot 2^2}. \] This simplifies further to: \[ L(\theta) = \frac{2\theta^3 (1-\theta)^2}{162}. \]

Step 2: Maximize the likelihood.
To find the maximum likelihood estimate, we take the derivative of \( L(\theta) \) with respect to \( \theta \) and set it equal to zero. First, we simplify the likelihood function: \[ L(\theta) = \frac{2\theta^3 (1-\theta)^2}{162}. \] Differentiate: \[ \frac{d}{d\theta} \left( 2\theta^3 (1-\theta)^2 \right) = 6\theta^2 (1-\theta)^2 - 4\theta^3 (1-\theta). \] Set the derivative equal to zero and solve for \( \theta \). After solving, we find that \( \theta = \frac{3}{5} \).

Step 3: Conclusion.
The maximum likelihood estimate of \( \theta \) is \( \frac{3}{5} \), so the correct answer is (B).