Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
Methods of Mathematical Statistics
4 Bivariate Distributions
Tim Brown and Guoqi Qian
Methods of Math. Stats.: Bivariate
1/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
Where have we come from?
PMF/PDF
For any real
x
the
Probability Mass/Density Function
f
(
x
)
, is given by
f
(
x
) =
P
(
X
=
x
)
,
X discrete
F
(
x
)
,
X continuous, F the CDF
(1)
Methods of Math. Stats.: Bivariate
2/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
Where have we come from?
PMF/PDF
For any real
x
the
Probability Mass/Density Function
f
(
x
)
, is given by
f
(
x
) =
P
(
X
=
x
)
,
X discrete
F
(
x
)
,
X continuous, F the CDF
(1)
Properties
(a)
0
≤
f
(
x
)(
and
≤
1
for
X
discrete
)
and
f
(
x
)
>
0
on the possible values of
X
(b)
∑
x
∈
range
(
X
)
f
(
x
) = 1
,
X discrete
∞
-∞
f
(
x
)
dx
= 1
,
X continuous
(c)
for any subset
A
of reals
P
(
X
∈
A
) =
∑
x
∈
A
f
(
x
)
,
X discrete
A
f
(
x
)
dx,
X continuous
Methods of Math. Stats.: Bivariate
2/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
Where are we going?
Consider
two random variables
X, Y
rather than one.
Methods of Math. Stats.: Bivariate
3/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
Where are we going?
Consider
two random variables
X, Y
rather than one.
Assume
both are discrete or both are continuous.
Methods of Math. Stats.: Bivariate
3/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
Where are we going?
Consider
two random variables
X, Y
rather than one.
Assume
both are discrete or both are continuous.
Base
our calculations to do with
X, Y
together on a
joint
PMF/PDF.
Methods of Math. Stats.: Bivariate
3/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
Where are we going?
Consider
two random variables
X, Y
rather than one.
Assume
both are discrete or both are continuous.
Base
our calculations to do with
X, Y
together on a
joint
PMF/PDF.
Aim:
Extend our capability to deal with
independent
RVs
(Variance, MGF) to
dependent
RVs.
Methods of Math. Stats.: Bivariate
3/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
Where are we going?
Consider
two random variables
X, Y
rather than one.
Assume
both are discrete or both are continuous.
Base
our calculations to do with
X, Y
together on a
joint
PMF/PDF.
Aim:
Extend our capability to deal with
independent
RVs
(Variance, MGF) to
dependent
RVs.
Need
to consider
joint
probabilities for
X, Y
.
Methods of Math. Stats.: Bivariate
3/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
New Notation for two random variables
X, Y
Write
for any sets
A, B
of real numbers the event
[
X
∈
A
]
∩
[
Y
∈
B
]
as
[
X
∈
A, Y
∈
B
]
Methods of Math. Stats.: Bivariate
4/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4
New Notation for two random variables
X, Y
Write
for any sets
A, B
of real numbers the event
[
X
∈
A
]
∩
[
Y
∈
B
]
as
[
X
∈
A, Y
∈
B
]
Example
[
X
= 1]
∩
[
Y
= 2]
is written as
[
X
= 1
, Y
= 2]
Methods of Math. Stats.: Bivariate
4/51

Joint, Marginal— 4.1,4.3
Cov, Corr’n — 4.2
Conditional Ds’ns - 4.3
Bivariate Normal — 4.4

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