Chapter 6
Continuous Random Variables
A continuous random variable can take any numerical value in some
interval. Assigning probabilities to individual values is not possible.
Probabilities can be measured in a given range.
For a continuous random variable
X
with a numerical value of
interest
x
the
cumulative distribution function
(
CDF
) is denoted
by:
with
)
x
X
(
P
)
x
X
(
P
)
x
(
F
<
=
≤
=
0
)
x
X
(
P
=
=
For two numerical values
a
and
b
, with
a < b
, the probability that
the outcome is in a range is:
)
a
(
F
)
b
(
F
)
a
X
(
P
)
b
X
(
P
)
b
X
a
(
P
)
b
X
a
(
P
−
=
<
−
<
=
≤
≤
=
<
<
Chapter 6
1

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The
probability density function
(
PDF
) is given by:
for all values of
x
.
0
)
x
(
f
>
The properties of a probability density function can be illustrated
with a special distribution called the
uniform distribution
.
The uniform distribution over the interval [0, 1] has the PDF:
⎪
⎩
⎪
⎨
⎧
<
<
=
otherwise
1
x
0
for
0
1
)
x
(
f
A graph of the probability density function is below.
1
0
1
a
0
f(x)
x
Area = P(X < a) = F(a)
Chapter 6
2