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CHAPTER 4
Continuous Random Variables and Probability Distributions
z
Basic deﬁnitions and properties of continuous random variables
Continuous random variable:
A random variable is continuous if its set of possible values
is an entire interval of numbers.
Probability distribution for continuous variables:
It is possible to construct a prob
ability histogram (same as relative frequency histogram) for continuous variable. But by
measuring the variable more and more ﬁnely, the resulting histogram approaches to a smooth
curve. It is obvious that the total area under this curve is 1, also probability that the variable
be between two points is the area under the curve between two points
. It means probabil
ity distribution or
probability density function (pdf)
for a continuous random variable
X
is a function
f
(
x
), such that
P
(
a
6
X
6
b
) =
Z
b
a
f
(
x
)
dx
f
(
x
) is a pdf if satisﬁes the following two conditions:
•
f
(
x
)
>
0 for all
x
•
R
∞
∞
f
(
x
)
dx
= 1 = area under the entire curve of
f
(
x
)
———————————————————————————
Example:
Suppose that
X
has following density function
f
(
x
) =
(
0
.
5
x
0
6
x
6
2
0
otherwise.
Calculate
a.
P
(
X
6
1)
b.
P
(0
.
5
6
X
6
1
.
5)
c.
P
(1
.
5
6
X
)
—————————————————————————–
Note:
For a continuous random variable
P
(
X
=
C
) = 0, then
P
(
a
6
X
6
b
) =
P
(
a < X
6
b
) =
P
(
a
6
X < b
) =
P
(
a < X < b
)
——————————————————————————–
Example:
Consider the following function
f
(
x
) =
(
0
.
15
e

0
.
15(
x

0
.
5)
x
>
0
.
5
0
otherwise.
1
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View Full Documenta.
Verify that
f
(
x
) is a pdf
b.
P
(
X
6
5)
———————————————————————————
z
Uniform Distribution
A continuous random variable
X
has uniform distribution on interval [
a,b
] if the pdf of
X
is
f
(
x
;
a,b
) =
(
1
b

a
a
6
X
6
b
0
otherwise.
z
Cumulative Distribution Function
The cumulative distribution function (pdf) for a continuous random variable
is
F
(
x
) =
P
(
X
6
x
) =
Z
x
∞
f
(
y
)
dy
∀
x
F
(
x
) is the area under the density curve to the left of
x
.
———————————————————
Example:
Find cdf for uniform distribution on [
a,b
] and then graph it.
————————————————————
Same as discrete random variable, the probabilities of intervals can be com
puted from
F
(
x
) as
P
(
X >
0) = 1

F
(
a
)
,
P
(
a
6
X
6
b
) =
F
(
b
)

F
(
a
)
—————————————————————————–
Example:
Solve example 1 by using cdf.
—————————————————————————–
•
If
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 Spring '09
 JHONBRAN

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