Properties (Properties of CDF)
•
If
y
≥
x
, then
F
X
(
y
) ≥
F
X
(
x
)
.
•
lim
x
→−∞
F
X
(
x
) =
0.
•
lim
x
→∞
F
X
(
x
) =
1.
Definition (Normal/Gaussian random variable)
A Normal random
variable
X
with mean
μ
and variance
σ
2
>
0 (
X
∼ N (
μ, σ
2
)
) has
PDF
f
X
(
x
) =
1
√
2
πσ
2
e
−
1
2
σ
2
(
x
−
μ
)
2
.
We have
E
[
X
] =
μ
and Var
(
X
) =
σ
2
.
Remark (Standard Normal)
The standard Normal is
N (
0
,
1
)
.
Proposition (Linearity of Gaussians)
Given
X
∼ N (
μ, σ
2
)
, and if
a
≠
0, then
aX
+
b
∼ N (
aμ
+
b, a
2
σ
2
)
.
Using this
Y
=
X
−
μ
σ
is a standard gaussian.
Conditioning on an event, and multiple continuous r.v.
Definition (Conditional PDF given an event)
Given a continuous
random variable
X
and event
A
with
P
(
A
) >
0, we define the
conditional PDF as the function that satisfies
P
(
X
∈
B
∣
A
) =
∫
B
f
X
∣
A
(
x
)
d
x.
Definition (Conditional PDF given
X
∈
A
)
Given a continuous
random variable
X
and an
A
⊂
R
, with
P
(
A
) >
0:
f
X
∣
X
∈
A
(
x
) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
P
(
A
)
f
X
(
x
)
,
x
∈
A,
0
,
x
/∈
A.
Definition (Conditional expectation)
Given a continuous random
variable
X
and an event
A
, with
P
(
A
) >
0:
E
[
X
∣
A
] =
∫
∞
−∞
f
X
∣
A
(
x
)
d
x.
Definition (Memorylessness of the exponential random variable)
When we condition an exponential random variable
X
on the event
X
>
t
we have memorylessness, meaning that the “remaining time”
X
−
t
given that
X
>
t
is also geometric with the same parameter
i.e.,
P
(
X
−
t
>
x
∣
X
>
t
) =
P
(
X
>
x
)
.
Theorem (Total probability and expectation theorems)
Given a
partition of the space into disjoint events
A
1
, A
2
, . . . , A
n
such that
∑
i
P
(
A
i
) =
1 we have the following:
F
X
(
x
) =
P
(
A
1
)
F
X
∣
A
1
(
x
) + ⋯ +
P
(
A
n
)
F
X
∣
A
n
(
x
)
,
f
X
(
x
) =
P
(
A
1
)
f
X
∣
A
1
(
x
) + ⋯ +
P
(
A
n
)
f
X
∣
A
n
(
x
)
,
E
[
X
] =
P
(
A
1
)
E
[
X
∣
A
1
] + ⋯ +
P
(
A
n
)
E
[
X
∣
A
n
]
.
Definition (Jointly continuous random variables)
A pair
(collection) of random variables is jointly continuous if there exists
a joint PDF
f
X,Y
that describes them, that is, for every set
B
⊂
R
n
P
((
X, Y
) ∈
B
) =
∬
B
f
X,Y
(
x, y
)
d
x
d
y.
Properties (Properties of joint PDFs)
•
f
X
(
x
) =
∞
∫
−∞
f
X,Y
(
x, y
)
d
y
.
•
F
X,Y
(
x, y
) =
P
(
X
≤
x, Y
≤
y
) =
x
∫
−∞
[
y
∫
−∞
f
X,Y
(
u, v
)
d
v
]
d
u
.
•
f
X,Y
(
x
) =
∂
2
F
X,Y
(
x,y
)
∂x ∂y
.
Example (Uniform joint PDF on a set
S
)
Let
S
⊂
R
2
with area
s
>
0, then the random variable
(
X, Y
)
is uniform over
S
if it has
PDF
f
X,Y
(
x, y
) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
s
,
(
x, y
) ∈
S,
0
,
(
x, y
) /∈
S.
Conditioning on a random variable, independence, Bayes’ rule
Definition (Conditional PDF given another random variable)
Given jointly continuous random variables
X, Y
and a value
y
such
that
f
Y
(
y
) >
0, we define the conditional PDF as
f
X
∣
Y
(
x
∣
y
)
△
=
f
X,Y
(
x, y
)
f
Y
(
y
)
.
Additionally we define
P
(
X
∈
A
∣
Y
=
y
)
∫
A
f
X
∣
Y
(
x
∣
y
)
d
x
.
Proposition (Multiplication rule)
Given jointly continuous
random variables
X, Y
, whenever possible we have
f
X,Y
(
x, y
) =
f
X
(
x
)
f
Y
∣
X
(
y
∣
x
) =
f
Y
(
y
)
f
X
∣
Y
(
x
∣
y
)
.
Definition (Conditional expectation)
Given jointly continuous
random variables
X, Y
, and
y
such that
f
Y
(
y
) >
0, we define the
conditional expected value as
E
[
X
∣
Y
=
y
] =
∫
∞
−∞
xf
X
∣
Y
(
x
∣
y
)
d
x.
Additionally we have
E
[
g
(
X
)∣
Y
=
y
] =
∫
∞
−∞
g
(
x
)
f
X
∣
Y
(
x
∣
y
)
d
x.

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- Fall '15
- Probability theory, random variable X