Consider a distribution with probability mass function 

where θ ∈ (0, 1) is an unknown parameter. In a random sample of size 100 from the above distribution, the observed counts of 0,1 and 2 are 20, 30 and 50 respectively. Then, the maximum likelihood estimate of θ based on the observed data is  

Concept: 

Maximum Likelihood Estimation (MLE), which is a method for estimating the parameters of a statistical model given observed data.

The likelihood function is the product of the probabilities of each observed outcome.

Explanation:

where  is an unknown parameter.

The sample size is 100.

The observed counts of x = 0, 1, 2 are 20, 30, and 50, respectively.

Let's denote the observed counts as

 for 

 for 

 for 

The likelihood function is the product of the probabilities of the observed data points, based on the PMF.

The probability of observing x = 0, 1, 2 given   is

The constant terms involving  can be ignored when maximizing the function.

To find the MLE, take the derivative of  with respect to  and set it equal to zero

For maximum value: 

⇒ 

Solving the equation: 

⇒ 

⇒ 

⇒ 

Substitute  and 

So, the correct option  is 2).