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January 21st, 2018, 08:42 AM   #1
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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?
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January 21st, 2018, 08:55 AM   #2
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Well, how do we define the bias? What happens if you compute it?
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January 21st, 2018, 11:04 AM   #3
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$E\left[\dfrac 1 n \sum \limits_{i=1}^n~X_i \right] = \dfrac 1 n \sum \limits_{i=1}^n~E[X_i] = \dfrac \lambda n \sum \limits_{i=1}^n~w_i$

can you finish it?
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