Applied Stochastic Processes Problem Set 3

Compute the Chernoff bound on P [X ≥ a] where X is a random variable that satisfies the exponential law f X (x) = λe −λx u(x). Recall, from Equation 4.6-4 on page 214 in [3], that the Chernoff bound for a continuous random variable X is given by P [X ≥ a] ≤ argmin t>0 e −at θ X (t) , (1) where θ X (t) is the moment-generating function θ X (t) E[e tX ] = ∞ −∞ e tx f X (x)dx (2) as defined by Equation 4.5-1 on page 211. Let's begin by finding the moment-generating function for an exponential random variable X. Substituting for f X (x) in Equation 2 gives the following expression. θ X (t) = λ ∞ 0 e tx e −λx dx = λ λ − t , for 0 < t < λ (3) Note that the integral in Equation 3 has a finite value only if t < λ and, since t > 0, we conclude 0 < t < λ. Substituting this result into Equation 1 gives the following form for the Chernoff bound. P [X ≥ a] ≤ argmin 0 0). P [X ≥ a] ≤ aλe 1−aλ , for a > 1 λ 1, for 0 < a ≤ 1 λ