Page 51 - Probability and Statistical Inference

28 1. Notions of Probability

1.6.2 The Median of a Distribution

Often a pdf has many interesting characteristics. One interesting characteris-
tic is the position of the median of the distribution. In statistics, when com-
paring two income distributions, for example, one may simply look at the
median incomes from these two distributions and compare them. Consider a
continuous random variable X whose df is given by F(x). We say that x is the
m
median of the distribution if and only if F(x ) = 1/2, that is P(X ≤ x ) = P(X ≥
m m
x ) = 1/2. In other words, the median of a distribution is that value x of X
m m
such that 50% of the possible values of X are below x and 50% of the
m
possible values of X are above x .
m
Example 1.6.8 (Example 1.6.4 Continued) Reconsider the pdf f(w) from
(1.6.7) and the df F(w) from (1.6.8) for the random variable W. The median w
m
of this distribution would be the solution of the equation F(w ) = 1/2. In view
m
1/3
of (1.6.8) we can solve this equation exactly to obtain u = (4.5) ≈ 1.651. !
m
1.6.3 Selected Reviews from Mathematics

A review of some useful results from both the differential and integral calculus
as well as algebra and related areas are provided here. These are not laid out in
the order of importance. Elaborate discussions and proofs are not included.
Finite Sums of Powers of Positive Integers:

Infinite Sums of Reciprocal Powers of Positive Integers: Let us write

Then, ζ (p) = ∞ if p ≤ 1, but ζ(p) is finite if p > 1. It is known that ζ(2) =
1/6π , ζ(4) = 1/90π , and ζ(3) ≈ 1.2020569. Refer to Abramowitz and Stegun
2
4
(1972, pp. 807-811) for related tables.