Random variables and distribution functions

2 free worked examples

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1. Exam PRandom variables and distribution functionsCore

   A loss $X$ has CDF $F(x)=1-(1+x{)}^{-3}$ for $x>0$. Find $P(1<X\le 4)$.

   1. A$0.0080$
   2. B$0.1170$
   3. C$0.1250$
   4. D$0.8750$
   5. E$0.9920$

   Solution

   1. Any interval probability is a difference of CDF values: $P(a<X\le b)=F(b)-F(a)$.
   2. $F(4)=1-5^{-3}=1-0.008=0.992$.
   3. $F(1)=1-2^{-3}=1-0.125=0.875$.
   4. $P=0.992-0.875=0.117$, so about 11.7% of losses fall in that band.
   5. Because the variable is continuous, using strict or weak inequalities makes no difference to the answer.

   Trap. Integrating the density from scratch when the CDF is already given - slower and an easy place to lose a constant.

2. Exam PRandom variables and distribution functionsExam level

   A random variable is uniform on the integers 1 through 20. Find $P(X>14)$.

   1. A$0.0500$
   2. B$0.2500$
   3. C$0.3000$
   4. D$0.3500$
   5. E$0.7000$

   Solution

   1. Each of the 20 integers has probability $1/20$.
   2. 'Greater than 14' means the values 15 through 20 - six of them.
   3. $P(X>14)=6/20$.
   4. $=0.30$. Counting carefully matters: 'at least 14' would give seven values and 0.35.

   Trap. Including 14 itself and answering 0.35.