Let Y be a Poisson random variable with mean . a. For a constant c > 0, show that E[log(Y + c)] = log

Let Y be a Poisson random variable with mean µ.

a. For a constant c > 0, show that

E[log(Y + c)] = log µ + (c – 1/2)/µ + O(µ^–2)

(Note that log(Y + c) = log µ + log[1 + (Y + c – µ)/µ].)

b. Cell counts in a 2 × 2 table are independent Poisson random variables. Use part (a) to argue that to reduce bias in estimating the log odds ratio, a sensible estimator is the sample log odds ratio after adding 1/2 to each cell.