Summary
Summary Intro to probability distributions and related concepts
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This material covers introductory statistical concepts, such as bivariate distributions and moment-generating functions.
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Whatis a multivariate probabilitydistribution?
Probabilities involving multiple with
leach
events a distribution occurring together, i.e.
the probability ofthe intersection of events.
↳
For example:P ( Y, y,) and P(Yc
=
yc)
=
and ... and PLYn yn)
=
P(X, y,,yz
=
= =
y2, ..., Yu yn)
=
or more compactly, P(y,y,,..., yn).
* joint probability
distribution.
It'slike calculating multiple individual probabilities simultaneously.
Where would multivariate distributions be used?
When we wantto investigate thatconsist
events ofmultiple individual events
occuring together.
the marginal distributions
given
How do we find the jointdistribution function?Does indepence ofthe random
1
variables
play a role?
In
general, you can't find the jointdistribution from the
marginal distributions,
unless thatthe variables independent, in which
you are
given are case
fxy(x,y) fx(x).fy(y).
=
For
jointdistributions of dependentvariables, see copulas and solar's theorem
, 40,
EX 5.14
by,"y,, 0
y,2y2,y, y22
f(y,,y2)
+
=
elsewhere.
joint density function, ssfleieldyet
(a)To be a valid
Ye 1
:"
Sy-sysyndy,dy, o.
giydyzdy, =
0
(yi (702)?") dy,
6(yp[z(z y,) - 1(3)"]dy,
=
-
6.y?(z(4
=
-
4y, y,)
+ -
typ]dy,
iyi (z
=
-
zy, +
typ -
zyp]dy,
=
6(2y, -
2ypdy,
12fyi y3dy,
=
-
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