Question

The number of minutes between successive bus arrivals at Mehmet's bus stop is exponentially distributed with unknown parameter .

Mehmet's prior information regarding  (real valued, i.e., a continuous random variable) can be summarized as a continuous uniform PDF in the range from 0 to 0.57.

Mehmet has recorded his bus waiting times for three days. These are 26, 11, and 25 minutes. Assume that Mehmet's bus waiting times in different days are independent.

Compute the MAP (maximum a posteriori probability) rule based estimate for  given the data collected by Mehmet.

Round your answer to five decimal digits after the decimal point.

Answer 1

Answer:-

Given that:-

Please go through the solution carefully specially the notations. You may consult any book or internet for further assistance regarding MAP estimation.

Hope the solution helps. Thank you.

Let X be the number of minutes between successive bus arrivals at minutes's

bus stop.

It is given that exponential

[As the from of the distribution is not given , we assume the PDF of X to be  

Also we know that

Thus  

we are required to find the MAP estimator of based on 3 independent observations on X. Let those be

The Joint pdf of the sample is

  

Now for MAP estination , we maximize the likelihood, which is defined as

  

Now,

  

Thus is in when  

attains maximum at decreasses with when

Now based on data,

  =20.67

=0.04837929366<0.57

Thus rounding off to 5 decimal digits after decimal point, the required estmate is =0.04838 minute.