Paper 2, Section I, F

Probability | Part IA, 2013

(i) Let XX be a random variable. Use Markov's inequality to show that

P(Xk)E(etX)ekt\mathbb{P}(X \geqslant k) \leqslant \mathbb{E}\left(e^{t X}\right) e^{-k t}

for all t0t \geqslant 0 and real kk.

(ii) Calculate E(etX)\mathbb{E}\left(e^{t X}\right) in the case where XX is a Poisson random variable with parameter λ=1\lambda=1. Using the inequality from part (i) with a suitable choice of tt, prove that

1k!+1(k+1)!+1(k+2)!+(ek)k\frac{1}{k !}+\frac{1}{(k+1) !}+\frac{1}{(k+2) !}+\ldots \leqslant\left(\frac{e}{k}\right)^{k}

for all k1k \geqslant 1.

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