Hypergeometric
. Set
P
(
X
=
i
) =
m
i
N

m
n

i
N
n
.
This comes up in sampling without replacement: if there are
N
balls, of which
m
are one color and the other
N

m
are another, and we choose
n
balls at random without replacement, then
X
represents the probability
of having
i
balls of the first color.
6. Continuous distributions.
A r.v.
X
is said to have a continuous distribution if there exists a nonnegative function
f
such that
P
(
a
≤
X
≤
b
) =
b
a
f
(
x
)
dx
for every
a
and
b
. (More precisely, such an
X
is said to have an absolutely continuous distribution.)
f
is
called the density function for
X
. Note
∞
∞
f
(
x
)
dx
=
P
(
∞
< X <
∞
) = 1. In particular,
P
(
X
=
a
) =
a
a
f
(
x
)
dx
= 0 for every
a
.
Example.
Suppose we are given
f
(
x
) =
c/x
3
for
x
≥
1. Since
∞
∞
f
(
x
)
dx
= 1 and
c
∞
∞
f
(
x
)
dx
=
c
∞
1
1
x
3
dx
=
c
2
,
we have
c
= 2.
Define
F
(
y
) =
P
(
∞
< X
≤
y
) =
y
∞
f
(
x
)
dx
.
F
is called the distribution function of
X
.
We
can define
F
for any random variable, not just continuous ones, by setting
F
(
y
) =
P
(
X
≤
y
). In the case
of discrete random variables, this is not particularly useful, although it does serve to unify discrete and
continuous random variables. In the continuous case, the fundamental theorem of calculus tells us, provided
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 Spring '09
 wen
 Probability distribution, Probability theory, probability density function, dx, absolutely continuous distribution.

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