Paper 2, Section II, 10F

Probability | Part IA, 2017

(a) For any random variable XX and λ>0\lambda>0 and t>0t>0, show that

P(X>t)E(eλX)eλt\mathbb{P}(X>t) \leqslant \mathbb{E}\left(e^{\lambda X}\right) e^{-\lambda t}

For a standard normal random variable XX, compute E(eλX)\mathbb{E}\left(e^{\lambda X}\right) and deduce that

P(X>t)e12t2\mathbb{P}(X>t) \leqslant e^{-\frac{1}{2} t^{2}}

(b) Let μ,λ>0,μλ\mu, \lambda>0, \mu \neq \lambda. For independent random variables XX and YY with distributions Exp(λ)\operatorname{Exp}(\lambda) and Exp(μ)\operatorname{Exp}(\mu), respectively, compute the probability density functions of X+YX+Y and min{X,Y}\min \{X, Y\}.

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