Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Let X and Y denote the two amounts. Their common mean is one, so each has rate one.

fX(x)=e−x,Pr⁡(Y>t)=e−t,x,t≥0f_X(x)=e^{-x},\qquad \Pr(Y>t)=e^{-t},\qquad x,t\ge 0

Model

Model

First evaluate the direction in which Y exceeds twice X. The opposite direction has the same probability, and the two strict inequalities cannot occur together.

Pr⁡(Y>2X)=∫0∞Pr⁡(Y>2x)fX(x) dx\Pr(Y>2X)=\int_0^\infty \Pr(Y>2x)f_X(x)\,dx

Compute

Compute

Insert the exponential survival function and density, then use symmetry.

Pr⁡(Y>2X)=∫0∞e−2xe−x dx=∫0∞e−3x dx=13\Pr(Y>2X)=\int_0^\infty e^{-2x}e^{-x}\,dx=\int_0^\infty e^{-3x}\,dx=\frac13
Pr⁡(Y>2X or X>2Y)=2(13)=23\Pr(Y>2X\text{ or }X>2Y)=2\left(\frac13\right)=\frac23

Answer

Answer

The required probability is two-thirds.

23(E)\boxed{\frac23\quad\text{(E)}}