# unbiased estimators

• Mar 15th 2011, 10:35 PM
holly123
unbiased estimators
suppose that a random variable X has a geometric distribution (pq^x) for which the parameter p is unknown (0<p<1). Find a statistic that will be an unbiased estimator of 1/p

Any help on this would be greatly appreciated, thank you so much
• Mar 15th 2011, 10:38 PM
matheagle
Do you have a sample or just one observation?
You do realize E(X)=1/p depending on how you define your distribution
http://en.wikipedia.org/wiki/Geometric_distribution
So, in that setting X+1 is the answer
• Mar 15th 2011, 10:44 PM
holly123
Is there a way to prove X+1 is an unbiased estimator of 1/p? We usually write out proofs which I have not gotten the hang of yet
• Mar 15th 2011, 10:46 PM
matheagle
IF you are using $\displaystyle P(X=x)=pq^x$ for x=0,1,2,3...

Then E(X)=(1/p)-1, do you need a proof of that?

THEN E(X+1)=E(X)+1=(1/p)-1+1=1/p.
• Mar 15th 2011, 11:04 PM
matheagle
OR for $\displaystyle P(X=x)=pq^{x-1}$ for x=1,2,3...

then E(X)=1/p. IN that case X is the answer
But do you have a sample X_1, X_2,....? or just one X?
• Mar 16th 2011, 06:55 AM
holly123
i think it is a sample
• Mar 16th 2011, 07:10 AM
matheagle
Then use $\displaystyle \bar X+1$