Average estimation of capped data

Problem setting

--Measuring a number of times as a sensor --This time is known to follow an exponential distribution --Sensor malfunction may return less than the correct time. However, you can tell if something went wrong. --I want to know the average of the original time from the measured values

a formula

It may be obtained so that the probability of becoming the estimated value is maximized. Specifically, the following equation. However, in the case of a defect, it is considered that the upper limit has been reached.

$ Original average = \ frac {total of all data} {number of data-number of limits reached} $

Check with Python

python3

import numpy as np
np.random.seed(1)
n = 100000 #The number of data
a = np.random.exponential(3, n) #Original distribution
print('Average of the original distribution%.3f'%a.mean())
b = np.random.uniform(2, 10, n) #upper limit
c = np.min((a, b), 0) #Distribution with upper limit
nn = (c==b).sum() #Number reached the upper limit
print('Estimated mean of the original distribution%.3f'%(c.sum()/(n-nn)))
>>>
Mean of original distribution 2.996
Estimated mean of the original distribution 2.996

that's all