Question

Let -1 and 1 be the observed values of a random sample of size two from \( N(\theta, \theta) \) distribution. The maximum likelihood estimate of \( \theta \) is

• A. 0
• B. 2
• C. \( \frac{-\sqrt{5}-1}{2} \)
• D. \( \frac{\sqrt{5}-1}{2} \)

Hint:

To find the maximum likelihood estimate, write the likelihood function, take the log, differentiate, and solve for the parameter of interest.

Answer 1

The Correct Option is D

Step 1: Write the likelihood function.
The likelihood function for a normal distribution with mean \( \theta \) and variance \( \theta \) for a sample \( x_1 \) and \( x_2 \) is given by: \[ L(\theta) = \prod_{i=1}^2 \frac{1}{\sqrt{2\pi \theta}} \exp\left( -\frac{(x_i - \theta)^2}{2\theta} \right) \] Substitute \( x_1 = -1 \) and \( x_2 = 1 \) into the likelihood function. Step 2: Maximize the likelihood function.
Take the logarithm of the likelihood function and differentiate with respect to \( \theta \). Set the derivative equal to zero to find the value of \( \theta \). Step 3: Conclusion.
After solving, we find that the maximum likelihood estimate of \( \theta \) is 0. Thus, the correct answer is \( \boxed{0} \).