Please answer all parts of the following question:

Suppose a chemist has two independent measurements XA and X3 available to estimate an unknown concentration level C. Suppose further that X A ~ N {C 4) and X3 ~ N(C 2]. Notice then that 0.4 XA and C3- X3 are both point estimes ofthe unknown concentration level 0. (a) Suppose we take the average of these two point estimmes to estiniate C: CA + 0.8 2 . Is this an unbiased point estimate of C? Show your work. (b) Suppose you want to combine the estimates in a more exible way, perhaps to reduce the variance of the estimator. So, yOu form the new estimator a: aCA-Zl-bC'B1 where o and b are xed positive real numbers. What condition (equation) must a and b need to satisfy in order for this new estimator to be unbiased? (c) What kind of distribution (give the name, and the parameters) does C' have when a = 0.5 and b = 1.5