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 / ii

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- Solution ii

Solution

Compute all three probabilities. From the preceding part,

P (A) = e^{- 6}

. Integrating the joint probability density function over

u > v

gives

P (B) = \int_{0}^{\infty} \int_{v}^{\infty} 2 e^{- u - 2 v} d u d v = \int_{0}^{\infty} 2 e^{- 3 v} d v = \frac{2}{3} .

On

A \cap B

one has

v > 2

and

u > v

, so

P (A \cap B) = \int_{2}^{\infty} \int_{v}^{\infty} 2 e^{- u - 2 v} d u d v = \frac{2}{3} e^{- 6} = P (A) P (B) .

Therefore $A$ 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.

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