Consider a distribution with probability mass function

(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

Correct option is B

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.

Solution:

​where θ∈(0,1)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

n₀ =20, for x = 0

n₁ =30 for x = 1

n₂ =50 for x = 2

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 θ  is 5/7