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
?Stanley Chan 2022. All Rights Reserved.
1 / 26
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 ).
?Stanley Chan 2022. All Rights Reserved.
2 / 26
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).
?Stanley Chan 2022. All Rights Reserved.
3 / 26
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
?Stanley Chan 2022. All Rights Reserved.
4 / 26
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.
?Stanley Chan 2022. All Rights Reserved.
5 / 26
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.
?Stanley Chan 2022. All Rights Reserved.
6 / 26
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
?Stanley Chan 2022. All Rights Reserved.
7 / 26
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 .
?Stanley Chan 2022. All Rights Reserved.
8 / 26
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- mole to mole answer key isd 622
- q 01 302 303 1 23 4 5 7 8 9 0 111 3 4 3056 7 p 302 303 308
- u s department of transportation
- ch302 equations and constants to be found on exams ch302
- rtamm34a bas g 11 12 0 4 0 10 5 0 302 0 30 8 3 0 6 0 page
- ece 302 homework 1
- ece 302 lecture 5 1 joint pdf and cdf
- 09 series fast lead screws southco inc
- 6 302 feedback systems
- unit ii interpolation approximation weebly
Related searches
- pdf to cdf examples
- pdf and cdf in probability
- pdf to cdf example
- pdf and cdf calculator
- pdf vs cdf probability
- pdf to cdf calculator
- pdf to cdf stats
- pdf to cdf formula
- 1 or 3 2 0 5 374 374 168 1 1 default username and password
- 1 or 3 2 0 5 711 711 168 1 1 default username and password
- 1 or 3 2 0 5 693 693 168 1 1 default username and password
- 1 or 3 2 0 5 593 593 or 2dvchrbu 168 1 1 default username and password