Topic 1: Probability foundations
Brendan K. Beare
September 22, 2011
1
Mass, density and distribution functions
A
discrete
random variable
X
can be equal to any of a finite number of possible
values. The probabilities with which
X
takes on each of those possible values are
given by its
probability mass function
, or pmf, which we denote
f
X
. The probability
that
X
is equal to some fixed value
x
is equal to
f
X
(
x
):
P
(
X
=
x
) =
f
X
(
x
)
.
Since the values of
f
X
(
x
) for different
x
’s represent probabilities, we must have
f
X
(
x
)
≥
0 for all
x
, and the sum of the probabilities must equal one:
∑
x
f
X
(
x
) = 1.
A
continuous
random variable
X
is equal to any fixed value
x
with probability
zero. The random behavior of
X
is described by its
probability density function
, or
pdf, which we also denote
f
X
. Probability density and mass functions are not the
same thing. When
X
is a continuous random variable, the probability that
X
lies
in an interval (
a, b
) is equal to the area under its pdf between
a
and
b
:
P
(
a < X < b
) =
Z
b
a
f
X
(
x
)d
x.
Since the probability of a random variable lying in any interval (
a, b
) must be
nonnegative, every pdf must be nonnegative:
f
X
(
x
)
≥
0 for all
x
.
Also, the
probability of a random variable lying anywhere on the real line is equal to one, so
we must have
R
∞
∞
f
X
(
x
)d
x
= 1
We can describe the behavior of discrete or continuous random variables using
their
cumulative distribution function
, or cdf, which we denote
F
X
. The probability
that
X
is equal to or less than some fixed value
x
is equal to
F
X
(
x
):
P
(
X
≤
x
) =
F
X
(
x
)
.
Every cdf is an nondecreasing function that increases from zero to one as we move
rightward along the
x
axis. When
X
is discrete, the probability that it is equal to
1
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or less than
x
is equal to the sum of its pmf
f
X
over all values of the argument
that are equal to or less than
x
:
F
X
(
x
) =
X
y
≤
x
f
X
(
y
)
.
When
X
is continuous, the probability that it is equal to or less than
x
is equal to
the area under its pdf
f
X
to the left of
x
:
F
X
(
x
) =
Z
x
∞
f
X
(
y
)d
y.
If
F
X
is differentiable,
X
must be a continuous random variable, and its pdf
f
X
is
the derivative of
F
X
. If
F
X
is a step function, then
X
is a discrete random variable,
and its pmf
f
X
(
x
) is equal to zero if
F
X
is continuous at
x
, and equal to the size
of the discontinuity of
F
X
at
x
for those
x
where
F
X
is discontinuous.
2
Expected value and variance
The
expected value
of a discrete random variable
X
with pmf
f
X
is defined as
E
(
X
) =
X
x
xf
X
(
x
)
.
More generally, the expected value of
g
(
X
) for some real valued function
g
is defined
as
E
(
g
(
X
)) =
X
x
g
(
x
)
f
X
(
x
)
.
Similarly, the expected value of a continuous random variable
X
with pdf
f
X
is
defined as
E
(
X
) =
Z
∞
∞
xf
X
(
x
)d
x,
and the expected value of
g
(
X
) is defined as
E
(
g
(
X
)) =
Z
∞
∞
g
(
x
)
f
X
(
x
)d
x.
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 Spring '08
 Stohs
 Normal Distribution, Probability theory, probability density function

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