Download Notes 26 - Wharton Statistics

Statistics 510: Notes 26
Reading: Section 8.5 (The One Sided Chebyshev
Inequality)
Chebyshev’s Inequality: If X is a random variable with
2
finite mean  and variance  , then for any value a  0 ,
P{| X   | a} 
2
a2 .
Chebyshev’s Inequality provides a bound for a two-sided
probability – what if we are only interested in a one-sided
probability, i.e., P( X    a) or P( X    a) ?
We can obtain a sharper bound.
One-sided Chebyshev’s Inequality:
If X is a random variable with finite mean  and variance
 2 , then for any value a  0 ,
P{ X    a} 
2
 2  a2
2 .
P{ X    a}  2
  a2
Proof: We first consider the case   0 , i.e., X is a random
variable with mean 0. Let b  0 and note that
X  a is equivalent to X  b  a  b . Hence,
P ( X  a )  P ( X  b  a  b)
 P{( X  b)2  (a  b)2 }
1
where the inequality is obtained by noting since a  b  0 ,
2
2
we have that X  b  a  b implies ( X  b)  (a  b) .
Upon applying Markov’s inequality, the preceding yields
that
E[( X  b)2 ]  2  b2
P( X  a ) 

2
(a  b)
(a  b)2 .
 2  b2
2
The value of b that minimizes (a  b)2 is b   / a ,
yielding for a  0
P( X  a) 
2
 2  a2
for X with mean 0
(1.1)
Now suppose X has mean  , which might not equal 0,
2
and variance  Then X   and   X have mean 0 with
2
variance  and we can apply (1.1) to obtain
P( X    a) 
2
 2  a2
2
P(   X  a)  2
  a2
Rearranging these inequalities gives
P{ X    a} 
2
 2  a2
2 .
P{ X    a}  2
  a2
Example 1: The mean of a list of a million numbers is 10
2
and the mean of the squares of the numbers is 101. Find an
upper bound on how many of the entries in the list are 14 or
more.
In Notes 24, we used the two-sided Chebyshev’s Inequality
to find a bound of 62,500. How much better can we do
with the one-sided Chebyshev’s Inequality?
3
Follow-up courses to Stat 510:
Statistics courses
Stat 511: Mathematical statistics (e.g., efficient point
estimation, Neyman-Pearson theory of optimal hypothesis
testing and statistical methodology (e.g., regression and
ANOVA). Spring, MW 10:30-12.
Stat 512: Mathematical statistics. Compared to 511, 512
goes more into depth into mathematical statistics and has
less focus on statistical methodology. Spring, TTh 3:004:30.
Probability courses:
Stat 530: Probability theory. Rigorous mathematical course
that would prove in generality and rigor the results we
proved here without focusing on applications, e.g., measure
theory would be discussed which can deal with continuous
random variables whose distribution is not smooth enough
to have a density and the central limit theorem would be
proved without assuming that a random variable has a
moment generating function using characteristic functions.
Offered in the fall.
OPIM 930: Stochastic processes. Good follow-up to this
course for students wanting to learn more probability
models (e.g., Markov chains) at the level of mathematical
rigor of this course. Offered in the fall.
4