Math Help - bias/unbiased estimator

having trouble with question in bold, any help will be appreciated.. Thanks!

Just look for a function of that is unbiased for . A good start would be and then fix it up so that it's unbiased. Running through this really quick, I got



which you can easily verify is unbiased and consistent. It'll help you out in doing this if you first verify the following: for , under the parametrization of the gamma given,



for all r such that .

for consistency (weak or strong) use the weak or strong law of large numbers for the sample mean.