Geometric Sum of Exponential Random Variables. Suppose X 1 , X 2 ,... are independent exponentially distributed with rate λ, and ν is a random variable independent of the Xn with the geometric...

Geometric Sum of Exponential Random Variables. Suppose X₁, X₂,... are independent exponentially distributed with rate λ, and ν is a random variable independent of the Xn with the geometric distribution P{N = n} = pⁿ⁻¹(1 − p), n ≥ 1. Prove that