 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2009/e/e3/b/i/solution

Write

r_{x} = d (x, a)

. The prescribed function is symmetric and nonnegative. If

x \neq = y

, both

r_{x}, r_{y}

could be zero only if

x = y = a

, so

d_{a} (x, y) > 0

. For the triangle inequality, if any two of

x, y, z

coincide it is immediate. If all three are distinct,

d_{a} (x, z) = r_{x} + r_{z} \leq (r_{x} + r_{y}) + (r_{y} + r_{z}) = d_{a} (x, y) + d_{a} (y, z) .

Thus

d_{a} is a metric

. Also

d_{a} (a, x) = d (a, x)

, a fact useful in the following parts.

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