1
This version was posted after
class.
WEEK 7 – Wed, Oct 13
Chapter 3.
Random Variables and Probability Distributions on the Line
.
Discrete –
Probability mass function
.
p
(
x
)
Cumulative distribution function
.
x
y
y
p
x
X
P
x
F
)
(
)
(
)
(
Expectation
.
)
(
)
(
x
xp
X
E
called the mean
of
X
or simply the expected
value of
X
.
For any function
h
and
Y
=
h
(
X
),
Y
is a random variable with expectation
)
(
)
(
))
(
(
)
(
x
p
x
h
X
h
E
Y
E
.
Variance and Standard Deviation
.
)]
(
[
)
(
]
)
[(
)
(
2
2
2
2
2
2
x
p
x
X
E
X
E
X
V
)
(
)
(
X
V
X
SD
Please note
.
The units of measure for
E
(
X
) and
SD
(
X
) are the same as the units of
measure for
X
.
If
X
is measured in
feet
, then
E
(
X
) and
SD
(
X
) are in units of
feet
.
If
X
is in
units of ($1,000), then
E
(
X
) and
SD
(
X
) are in units of ($1,000).
The expectation of
X
, written
E
(
X
), is a measure of the “center
” of the probability
distribution.
It is the balance point for the distribution of probability masses positioned at
the possible values of
X
.
In the dice example, the plots of the distribution of
X
= Total on
Pair of Fair Dice show by symmetry that the balance point is 7 so that we can conclude
that
7
)
(
X
E
without computation.
For nonsymmetric distributions, we can only
guess the balance point from the plot.
The standard deviation of X, written
SD
(
X
), is a
measure of the “spread
” of the probability distribution.
We will build up intuition in
regard to this measure as we proceed further in the course.
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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)
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 Summer '10
 DennisGilliland
 Normal Distribution, Probability, Probability theory, probability density function, µ

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