Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 8 / b

Using the rate parameter convention for the exponential distribution, independence gives the joint probability density functionFor $z\ge 0$, the survival probability of the minimum isThus $Z$ has exponential distribution with rate $3$, andValues of a probability density function at the single endpoint $0$ do not affect the probability distribution.

Compute all three probabilities. From the preceding part, $\mathbb{P}(A)=e^{-6}$. Integrating the joint probability density function over $u>v$ givesOn $A\cap B$ one has $v>2$ and $u>v$, soTherefore $A$ and $B$ are independent events. The minimum waiting time and the identity of the first clock are independent for competing exponential clocks; unequal rates change the winner probabilities but do not destroy that independence.

Choose $g(t)=e^t$. If $W$ has an exponential distribution with rate $r$, then for $c<r$,while the integral diverges to $+\infty$ when $c\ge r$. Since $Z$ has rate $3$,For the other two rates, $\mathbb{E}[g(V)]=2/(2-1)=2$, whereas $\mathbb{E}[g(U)]={\int }_0^{\infty }1dt=+\infty$. These computations establish exactly the required finite variance, finite expected value, and infinite expected value.