ECE 302: Lecture 5.1 Joint PDF and CDF
[Pages:26]ECE 302: Lecture 5.1 Joint PDF and CDF
Prof Stanley Chan
School of Electrical and Computer Engineering Purdue University
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What are joint distributions?
Joint distributions are high-dimensional PDF (or PMF or CDF).
fX (x ) = fX1,X2 (x1, x2) = fX1,X2,X3 (x1, x2, x3)
one variable
two variables
three variables
= . . . = fX1,...,XN (x1, . . . , xN ).
N variables
Notation:
fX (x ) = fX1,...,XN (x1, . . . , xN ).
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Why study joint distributions?
Joint distributions are ubiquitous in modern data analysis. For example, an image from a dataset can be represented by a high-dimensional vector x. Each vector has certain probability to be present. Such probability is described by the high-dimensional joint PDF fX (x).
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Outline
Joint PDF and CDF Joint Expectation Conditional Distribution Conditional Expectation Sum of Two Random Variables Random Vectors High-dimensional Gaussians and Transformation Principal Component Analysis Today's lecture Joint PMF, PDF Joint CDF Marginal PDF Independence
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Joint PMF
Definition Let X and Y be two discrete random variables. The joint PMF of X and Y is defined as
pX ,Y (x, y ) = P[X = x and Y = y ].
(1)
Figure: A joint PMF for a pair of discrete random variables consists of an array of impulses. To measure the size of the event A, we sum all the impulses inside A.
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Example
Example 1. Let X be a coin flip, Y be a dice. Find the joint PMF.
Solution. The sample space of X is {0, 1}. The sample space of Y is {1, 2, 3, 4, 5, 6}. The joint PMF is
Y
123456
X=0
1 12
1 12
1 12
1 12
1 12
1 12
X=1
1 12
1 12
1 12
1 12
1 12
1 12
Or written in equation:
1
pX ,Y (x , y )
=
, 12
x = 0, 1,
y = 1, 2, 3, 4, 5, 6.
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Example
Example 2. In the previous example, define A = {X + Y = 3} and B = {min(X , Y ) = 1}. Find P[A] and P[B].
Solution:
P[A] =
pX ,Y (x , y ) = pX ,Y (0, 3) + pX ,Y (1, 2)
(x,y )A
2 =
12
P[B] =
pX ,Y (x , y )
(x,y )B
= pX ,Y (1, 1) + pX ,Y (1, 2) + . . . + pX ,Y (1, 5) + pX ,Y (1, 6) 6
=. 12
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Joint PDF
Definition Let X and Y be two continuous random variables. The joint PDF of X and Y is a function fX,Y (x, y ) that can be integrated to yield a probability:
P[A] = fX ,Y (x, y )dxdy ,
(2)
A
for any event A X ? Y .
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