Exercise 2.17 - Expected value of the minimum

Let $m$ denote the location of the left most point, we have:

Therefore:

For the second equation, note that:

therefore

There are perhaps more intuitive solutions to this problem, for example, plotting the value of $X$, $Y$ and $\min <!--\; FUNCTION\; APPLICATION\; -->(X,Y)$ into one graph:

The height of the cyan pyramid at $(x,y)$ marks the value of $\min <!--\; FUNCTION\; APPLICATION\; -->(x,y)$, so the expectation of the statistics equals the average height, in this case also the volume of the pyramid, $\frac{1}{3}$. One can also graphically compute the average distance between $X$ and $Y$ from the following plot:

Since the average distance between $X$ and $Y$ is $\frac{1}{3}$, so is that between $\min <!--\; FUNCTION\; APPLICATION\; -->(X,Y)$ and $\max <!--\; FUNCTION\; APPLICATION\; -->(X,Y)$. Moreover, we have $𝔼[\min <!--\; FUNCTION\; APPLICATION\; -->(X,Y)+\max <!--\; FUNCTION\; APPLICATION\; -->(X,Y)]=𝔼[X]+𝔼[Y]=1$, hence $\min <!--\; FUNCTION\; APPLICATION\; -->(X,Y)=\frac{1}{3}$.

However, the graphical method, although entertaining and inspiring, should not be considered as a reliable option in proving probability properties. The dependency can complicate the underlying topology (e.g., the $X-Y$ plane might have another geometry other than the Euclidean one, if $X$ and $Y$ are not independent), resulting in confusions and fallacies.

For example, if $X$ is subject to a uniform distribution on $[0,1]$ while $Y$ is uniformly distributed in $[\max <!--\; FUNCTION\; APPLICATION\; -->(0,X-0.2),\min <!--\; FUNCTION\; APPLICATION\; -->(1,X+0.2)]$. Then the pyramid in Fig. 1 is left with the region along the diagonal line in the $X$- $Y$ plane. This generalization is insignificant since it only changes the region where the integral shall be done.

Consider the case where $X$ and $Y$ are independent random variables subject to truncated Gaussian centered at $\frac{1}{2}$ on $[0,1]$. The plot for visualizing $\min <!--\; FUNCTION\; APPLICATION\; -->(X,Y)$ is the same as Fig. 1. However, to compute the expectation of $\min <!--\; FUNCTION\; APPLICATION\; -->(X,Y)$, one cannot simply calculate the volume of the pyramid since the geometry of the $X$- $Y$ plane has changed.