Chapter 4: Probability
Distributions
1
Random Variables
}
Random Variable
X associates a numerical value to each outcome of an
experiment.
(r.v.)
Example 1:
X: # heads obtained in three tosses of a fair coin.
Outcome
X
HHH
3
HHT
2
HTH
2
HTT
1
THH
2
THT
1
TTH
1
TTT
0
Value of X
Event
X = 0
{
TTT
}
X = 1
{
HTT, THT, TTH
}
X = 2
{
HHT, HTH, THH
}
X = 3
{
HHH
}
P
(
X
= 0) =
1
8
P
(
X
= 1) =
3
8
P
(
X
= 2) =
3
8
P
(
X
= 3) =
1
8
}
A r.v. is discrete
if its set of possible values is a discrete set.
}
A r.v. is continuous
if it represents some measurement on a continuous
scale.
eg)
X: # heads obtained in three tosses of a coin:
continuous.
X: amount of precipitation produced by a storm: discrete.
2
Probability distribution of a discrete r.v.
}
Probability distribution of a discrete r.v.:
f
(
χ
) =
P
(
X
=
χ
)
}
f
(
χ
)
≥
0
for all
χ
∑
allχ
f
(
χ
) = 1
Example 2:
1
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(a)
f
(
χ
) =
χ

2
2
for
χ
= 1
,
2
,
3
,
4
f
(1) =

1
2
⇒
not
a probability distribution.
(b)
f
(
χ
) =
χ
2
25
for
χ
= 0
,
1
,
2
,
3
,
4
f
(
χ
)
≥
0 for all
χ
, but
4
∑
χ
=0
f
(
χ
) = 0+
1
25
+
4
25
+
9
25
+
16
25
=
30
25
=
6
5
6
= 1
∴
not
a probability distribution.
Example 3:
30% of the trees in a forest are infested with a parasite.
Four trees are selected at random.
X: # trees sampled that have the parasite.
Let I: infested,
N: not infested.
X=0
X=1
X=2
X=3
X=4
NNNN
NNNI
NNII
NIII
IIII
NNIN
NINI
INII
NINN
NIIN
IINI
INNN
INNI
IIIN
ININ
IINN
P
(
X
= 0) =
P
(NNNN) = (0
.
7)
4
= 0
.
2401
P
(
X
= 1) = 4
·
(0
.
7)
3
·
(0
.
3) = 4
·
(0
.
1029) = 0
.
4116
P
(
X
= 2) = 6
·
(0
.
7)
2
·
(0
.
3)
2
= 0
.
2646
P
(
X
= 3) = 4
·
(0
.
7)
·
(0
.
3)
3
= 0
.
0756
P
(
X
= 4) = (0
.
3)
4
= 0
.
0081
0
1
2
3
4
0.2
0.4
f(X)
}
Probability model:
An assumed form of the probability distribution that
describes the chance behavior for a r.v. X.
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 Spring '08
 Mendell
 Probability distribution

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