Using the Chen-Stein method and probabilistic coupling, Barbour et al. [4] show that the statistic Wd satisfies the inequality sup A⊂N

|Pr(Wd ∈ A) − Pr(Z ∈ A)| ≤ 1 − e−λ

λ [λ − Var(Wd)],(4.9)

where Z is a Poisson random variable having the same expectation

λ = m i=1 µi as Wd, and where N denotes the set {0, 1 ...} of nonnegative integers. Prove that

λ − Var(Wd) =

i

µ2 i −

i

j=i Cov(1{Ni≥d}, 1{Nj≥d}).

In view of Problem 3, the random variables 1{Ni≥d} and 1{Nj≥d} are negatively correlated. It follows that the bound (4.9) is only useful when the number λ−1(1 − e−λ)



i µ2 i is small. What is the value of

λ−1(1 − e−λ)



i µ2 i for the hemoglobin data when d = 2? Careful estimates of the difference λ − Var(Wd) are provided in [4].