A point \(P\) lies in an equilateral
triangle \(ABC\) of height 1. The perpendicular
distances from \(P\) to the sides \(AB\), \(BC\) and \(CA\) are
\(x_1\), \(x_2\) and \(x_3\), respectively. By considering the
areas of triangles with one vertex at \(P\), show
that \(x_1+x_2+x_3=1\).
Suppose now that \(P\) is placed at random in the equilateral triangle
(so that the probability of it lying in any given region of the triangle is
proportional to the area of that region). The perpendicular
distances from \(P\) to the sides \(AB\), \(BC\) and \(CA\) are
random variables \(X_1\), \(X_2\) and \(X_3\), respectively.
In the case \(X_1= \min(X_1,X_2,X_3)\), give a sketch showing
the region of the triangle in which \(P\) lies.
Let \(X= \min(X_1,X_2,X_3)\). Show that
the probability density function for \(X\) is
given by
\[
\f(x) =
\begin{cases}
6(1-3x) & 0 \le x \le \frac13\,, \\
0 & \text{otherwise}\,.
\end{cases}
\]
Find the expected value of \(X\).
A point is chosen at random in a regular tetrahedron of height 1.
Find the expected value of the distance from the point to the
closest face.
\newline
[The volume of a tetrahedron is
\(\frac13 \times \text{area of base}\times\text{height}\) and its centroid
is a distance \(\frac14\times \text{height}\) from the base.]