Transformations and order statistics

2 free worked examples

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1. Exam PTransformations and order statisticsCore

   $U$ is uniform on $(0,1)$ and $Y=-2\ln U$. Find $P(Y>3)$.

   1. A$0.2231$
   2. B$0.3333$
   3. C$0.5000$
   4. D$0.6065$
   5. E$0.7769$

   Solution

   1. Work through the CDF: $Y>3⟺-2\ln U>3⟺\ln U<-1.5$.
   2. The inequality FLIPPED when dividing by $-2$ - the step most often missed.
   3. $\ln U<-1.5⟺U<e^{-1.5}$, and for a standard uniform $P(U<c)=c$.
   4. $P(Y>3)=e^{-1.5}=0.2231$. In fact $Y$ is exponential with mean 2, which is the standard way to simulate exponentials.

   Trap. Failing to reverse the inequality when dividing by a negative number.

2. Exam PTransformations and order statisticsExam level

   $Z$ is standard normal and $X=5Z+20$. Find $P(X>28)$.

   1. A$0.0548$
   2. B$0.1587$
   3. C$0.2119$
   4. D$0.7881$
   5. E$0.9452$

   Solution

   1. A linear transform of a normal is normal: $X$ has mean 20 and standard deviation 5.
   2. Standardise: $z=\frac{28-20}{5}=1.6$.
   3. $P(X>28)=1-\Phi (1.6)=1-0.945201$.
   4. $=0.054799$, so about one observation in eighteen exceeds 28.
   5. Recognising that a linear transform of a normal stays normal is what makes this a two-line problem rather than a change-of-variable exercise.