Expected value of second highest draw

#1
Hello all,

Suppose I take two draws from a uniform distribution on [0,1]. I know that the value of the highest draw is equal to x. What then is the expected value of the second highest draw conditional on the highest draw being x? What is the distribution of the second highest draw (conditional on the knowledge our highest draw is x)?

Thank you kindly for your help,

Nick
 

BGM

TS Contributor
#2
Let \( U_1, U_2 \sim {\rm Uniform}(0,1) \) be the random sample

Let \( X = U_{(2)} \geq U_{(1)} = Y \) be the ordered sample

Then \( \forall x \in (0, 1), \forall y \in (0, x],\)

\( F_{Y|X = x}(y|x) = \Pr\{Y \leq y|X = x\} \)

\( = \Pr\{Y \leq y|Y \leq x\} \)

\( = \frac {\Pr\{Y \leq y, Y \leq x\}} {\Pr\{Y \leq x\}} \)

\( = \frac {\Pr\{Y \leq y\}} {\Pr\{Y \leq x\}} \)

\( = \frac {y} {x} \)

\( \Rightarrow Y|X = x \sim {\rm Uniform}(0, x) \)