Transformations and order statistics

2 free worked examples

Solved in full, free to everyone, and never rotated. Read these before deciding whether the rest of the drill is worth paying for.

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: $Y>3 \iff -2\ln U > 3 \iff \ln U < -1.5$.
   2. The inequality FLIPPED when dividing by $-2$ - the step most often missed.
   3. $\ln U < -1.5 \iff U < e^{-1.5}$, 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: $z = \dfrac{28-20}{5} = 1.6$.
   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.