1
Continuous Random Variables
Reading: Chapter 3.1 – 3.8
Homework: 3.1.2, 3.2.1, 3.2.4, 3.3.2, 3.3.7
G. Qu
ENEE 324 Engineering Probability
2
Cumulative Distribution Function
CDF of a random variable X is: F
X
(x) = P[X≤x]
X is a continuous random variable is F
X
(x) is
continuous. (the range of X contains a
continuous interval)
Example:
X: a random integer between 0 and 4.
S
X
={0,1,2,3,4}
P
X
(x) = 0.2 for x=0,1,2,3,4 and 0 otherwise
F
X
(x) is an nondecreasing piecewise constant function that is not
continuous at x=0,1,2,3,4
Y: a random real number between 0 and 4.
S
Y
=[0,4] is a continuous region, not a countable set.
F
Y
[y] = P[Y≤y] = ?
P
Y
[Y=y] = ?
>
≤
≤
<
=
4
1
4
0
4
/
0
0
)
(
y
y
y
y
y
F
Y
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G. Qu
ENEE 324 Engineering Probability
3
Properties of CDF
Theorem 3.1: for any random variable X
F
X
(∞) = 0, F
X
(∞) = 1
F
X
(x) = 0 for x<x
min
, F
X
(x) = 1 for x ≥x
max
F
X
(x) ≥ F
X
(x’) if x≥x’
F
X
(x) – F
X
(x’) = P[x’< X ≤ x]
Example: Quiz 3.1
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 Spring '11
 Gu
 Probability theory, probability density function, Cumulative distribution function, G. Qu

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