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January 21st, 2018, 09:42 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 100 Thanks: 2  Probability Estimators question
Suppose the random variables $X_{1}...x_{n}$ are independent and that for i=1,2...n \[X_{i}~Poisson(w_{i}\lambda)\] for $w_{i}>0$ lambda>0 for some weights $w_{1}...w_{n}$ calculate the bias if the sample mean $X_{n}$ is used as an estimator for lambda. What constraint can be placed on the weights to ensure the estimator is unbiased? 
January 21st, 2018, 09:55 AM  #2 
Senior Member Joined: Oct 2009 Posts: 712 Thanks: 238 
Well, how do we define the bias? What happens if you compute it?

January 21st, 2018, 12:04 PM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,266 Thanks: 1198 
$E\left[\dfrac 1 n \sum \limits_{i=1}^n~X_i \right] = \dfrac 1 n \sum \limits_{i=1}^n~E[X_i] = \dfrac \lambda n \sum \limits_{i=1}^n~w_i$ can you finish it? 

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