For X ∼ Poisson(λ) how can we calculate the coverage probability, confidence set can be obtd using chi square but not the coverage probability?

Question

For X ∼ Poisson(λ), show that the coverage probability of the confidence interval [L(X),U(X)] is given by ∞ Pλ(λ ∈ [L(X),U(X)]) = x=0 IL(x),U(x)e−λλx x! and that we can define functions xl(λ)andxu(λ)sothat xu(λ) Pλ(λ ∈ [L(X),U(X)]) = x=xl(λ) e−λλx x! .