Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 2 / 10F (Probability)

Apply Markov's inequality to the nonnegative random variable $e^{\lambda X}$, since $X>t$ implies $e^{\lambda X}>e^{\lambda t}$:This exponential Markov bound remains true, though uninformative, when the expected value is infinite. For a standard normal distribution, completion of the square gives its moment-generating functionMinimize ${\lambda }^2/2-\lambda t$ over positive $\lambda$; the choice $\lambda =t$ gives the Chernoff bound

Use the rate convention for the exponential distribution. By convolution of independent random variables, the probability density function of $S=X+Y$ is, for $s\ge 0$,It is zero for $s<0$. The expression is nonnegative whichever rate is larger, and defines a two-stage hypoexponential distribution. As a check, its limit as $\mu \to \lambda$ is ${\lambda }^{2}s{e}^{-\lambda s}$, the shape-two gamma distribution.

For $M=\min \{X,Y\}$, independence gives the survival function $\mathbb{P}(M>s)=e^{-\lambda s}{e}^{-\mu s}$ for $s\ge 0$. Differentiating its complement givesThus the minimum has an exponential distribution whose rate is the sum of the rates.