Suppose that F i , i ∈ I, is a finite collection of exponential distributions with respective rates λ i , i ∈ I, that are distinct. For J ⊆ I, let FJ denote the convolution of the distributions F j ,...

Suppose that F_i, i ∈ I, is a finite collection of exponential distributions with respective rates λ_i, i ∈ I, that are distinct. For J ⊆ I, let FJ denote the convolution of the distributions F_j
, j ∈ J. Show that, for any real numbers ai, and subsets Ji of I, for j ∈ I,This assertion also holds when I is infinite, under the additional assumption that the summations exist. Hint: Use the fact that the Laplace transform of the left-hand side of (3.54) is