Chebyshev Inequality
.
where
k
is any real number
k
≥ 1.
Example 1.
So, if
µ
= 50 and
σ
= 5,
Example 2.
Suppose that
Y
is a random variable with
µ
=
E
(
Y
) = $20,000 and
σ
=
SD
(
Y
) =
$5,477.
Use Chebyshev to determine a lower bound for the probability
P
(|
Y
– $20,000| <
$10,000).
Ans.
Note that the event in question is $10,000 < Y < $30,000.
Recall that a
z
-score tells
you how many standard deviations a point is away from the mean; the
z
-score of $30,000
is ($30,000 - $20,000)/$5,477 = 1.826 and the
z
-score of $10,000 is ($10,000 - $20,000)/
$5,477 = -1.826.
Thus,
P($10,000 <
Y
< $30,000) = P($20,000 – 1.826($5,477) <
Y
< ($20,000 + 1.826($5,477))
0.70.
SPECIAL PROBABILITY DISTRIBUTIONS
A probability distribution is a model for the uncertainty of a random variable.
A random variable has as its outcomes a set of real numbers.
If the set of outcomes can be
listed in a table, the random variable is said to be discrete.
Continuous random variables
take values across a continuum of real numbers, perhaps, an interval.
Certain probability distributions arise so commonly in applications, that they are studied
extensively.
Examples:
•
BINOMIAL PROBABILITY DISTRIBUTIONS (DISCRETE)
•
GEOMETRIC DISTRIBUTIONS (DISCRETE)
•
HYPERGEOMETRIC PROBABILITY DISTRIBUTIONS (DISCRETE)
•
UNIFORM PROBABILITY DISTRIBUTIONS (CONTINUOUS)
•
EXPONENTIAL PROBABILITY DISTRIBUTIONS (CONTINUOUS)
•
NORMAL PROBABILITY DISTRIBUTIONS (CONTINUOUS)
For discrete random variables, the probabilities come in masses.
With
2