Problem 8.17 Easy Free 1 variation

Exponential Race After Scaling

An ellipse centred at the origin has horizontal radius $A$ and vertical radius $B$ , where $A,B$ are independent $\mathrm{Exp}(1)$ random variables. Recall that an ellipse with radii $a$ and $b$ has area $\pi ab$ . A circle is drawn whose diameter equals $A$ . Find the probability that the circle’s area is larger than the ellipse’s area.

Work it out, then compare with the answer.

SolutionShow the worked solution

The circle has radius $A/2$ , so its area is $C=\frac{\pi}{4}A^2$ , while the ellipse has area $E=\pi AB$ . Since $A>0$ almost surely, we may divide by $\pi A$ :

C>E \iff \tfrac14 A^2 > AB \iff A>4B .

A scaled exponential is again exponential. Indeed $\mathbb{P}(4B>t)=e^{-t/4}$ , so $4B\sim\mathrm{Exp}(\tfrac14)$ , and it is independent of $A\sim\mathrm{Exp}(1)$ .

The event $A>4B$ says that $4B$ wins the race against $A$ . For independent $\mathrm{Exp}(\lambda)$ and $\mathrm{Exp}(\mu)$ variables, the first is the smaller with probability $\lambda/(\lambda+\mu)$ . Therefore

\mathbb{P}(A>4B)=\frac{1/4}{1/4+1}=\frac15 .

\boxed{\tfrac15}

Key idea. Rescaling turns the comparison into an exponential race with rates $1$ and $\tfrac14$ .

Variations

The same idea with a twist. Each solution gives only the changed step.

Variation 8.17a

Let $X_1,X_2$ be i.i.d. $\mathrm{Exp}(1)$ random variables, with order statistics $O_1=\min(X_1,X_2)$ and $O_2=\max(X_1,X_2)$ . Find the probability that the range $O_2-O_1$ exceeds the midpoint $(X_1+X_2)/2$ .

Work it out, then compare with the answer.

Solution

Since $\{X_1,X_2\}=\{O_1,O_2\}$ , the midpoint equals $(O_1+O_2)/2$ . Write $G=O_2-O_1$ for the gap. Then

O_2-O_1>\tfrac12(O_1+O_2)\iff O_2>3O_1\iff G>2O_1 .

The minimum $O_1$ is $\mathrm{Exp}(2)$ . By memorylessness, the gap $G$ is $\mathrm{Exp}(1)$ and independent of $O_1$ .

Rescaling gives $2O_1\sim\mathrm{Exp}(1)$ . So $G$ and $2O_1$ are i.i.d. $\mathrm{Exp}(1)$ , and by symmetry each is the larger with probability one half:

\boxed{\tfrac12}

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