Question

The time intervals between two consecutive accidents in an intersection were recorded to be 3, 10, 2, 3, 3, 11, 7, and 5 days. Suppose the time interval is geometrically distributed. Determine its parameter using the method of moments, and using the method of maximum likelihood. Compare the results.

Answer 1

Sol:

a) .

For method of moment estimation of the estimator we know that:

From sample data we get: the sample mean is computed as:

Now the method of moment estimation of paramter p would be by equating the expected value of the distribution to the sample mean value. Therefore we get:

Therefore, the estimate for p would be:

b).

Maximum likelihood estimation for the parameter p

The likelihood function would be defined as:

Putting the value of sum of X_i in the above equation we get:

Taking natural log both sides, we get:

Differentiating the above equation with respect to p, we get:

Equating the first derivative to 0 we get:

Therefore we get the same estimation for the parameter p with both the methods.