Continuous distributions

2 free worked examples

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1. Exam PContinuous distributionsCore

   Claim sizes are exponential with mean 800. What is the probability a claim exceeds 1,200?

   1. A$0.2231$
   2. B$0.3679$
   3. C$0.4724$
   4. D$0.5276$
   5. E$0.6321$

   Solution

   1. For an exponential with mean $\theta$, the survival function is $S(x)=e^{-x/\theta }$ - the single most useful form of the distribution.
   2. Here $\theta =800$, so $S(1200)=e^{-1200/800}=e^{-1.5}$.
   3. $=0.223130$.
   4. The rate parameter is $\lambda =1/800$; mixing up rate and mean is the classic slip, and it would give $e^{-960000}$, an obviously absurd answer.

   Trap. Using λ = 800 instead of λ = 1/800.

2. Exam PContinuous distributionsExam level

   Losses are exponential with mean 500. Given a loss exceeds 300, what is the probability it exceeds 800?

   1. A$0.2019$
   2. B$0.3679$
   3. C$0.4493$
   4. D$0.5488$
   5. E$0.7408$

   Solution

   1. The exponential is memoryless: $P(X>s+t\mid X>s)=P(X>t)$.
   2. Here $s=300$ and $s+t=800$, so $t=500$.
   3. $P(X>500)=e^{-500/500}=e^{-1}$.
   4. $=0.367879$. Computing $S(800)/S(300)=e^{-1.6}/e^{-0.6}$ gives the same thing - which IS the memoryless property.

   Trap. Reporting the unconditional S(800) = 0.2019 and ignoring the information already given.