{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec5print

# lec5print - Jointly distributed random variables Joint...

This preview shows pages 1–11. Sign up to view the full content.

Jointly distributed random variables Joint distributions and the central limit theorem Sayan Mukherjee Sta. 113 Chapter 5 of Devore September 27, 2007 Sayan Mukherjee Joint distributions and the central limit theorem

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Jointly distributed random variables Table of contents 1 Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Sayan Mukherjee Joint distributions and the central limit theorem
Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Joint distributions Definition Let X , Y be two discrete random variables. The joint pdf p ( x , y ) is defined by p ( x , y ) = IP ( X = x and y = Y ) , and for a set A IP [( x , y ) A ] = X x , y A p ( x , y ) . Sayan Mukherjee Joint distributions and the central limit theorem

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example An evil Leprechaun is chopping of fingers and toes at night. People wake up with X = 1 , 2 fingers and Y = 2 , 3 , 4 toes. Sayan Mukherjee Joint distributions and the central limit theorem
Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example p ( x , y ) y = 2 y = 3 y = 4 x = 1 . 2 . 1 . 2 x = 2 . 05 . 15 . 3 IP ( y > 2) = p ( x = 1 , y = 3) + p ( x = 1 , y = 4) + p ( x = 2 , y = 3) + p ( x = 2 , y = 4) = . 1 + . 2 + . 15 + . 3 = . 75 IP ( x = 2 , y < 4) = p ( x = 2 , y = 3) + p ( x = 2 , y = 2) = . 15 + . 05 = . 2 Sayan Mukherjee Joint distributions and the central limit theorem

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Marginal distributions Definition The marginal distributions of p ( x , y ) denoted by p X ( x ) and p Y ( y ) are given by p X ( x ) = X y p ( x , y ) p Y ( y ) = X x p ( x , y ) . Sayan Mukherjee Joint distributions and the central limit theorem
Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example p ( x , y ) y = 2 y = 3 y = 4 x = 1 . 2 . 1 . 2 x = 2 . 05 . 15 . 3 Sayan Mukherjee Joint distributions and the central limit theorem

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example p ( x , y ) y = 2 y = 3 y = 4 p X ( x ) x = 1 . 2 . 1 . 2 .5 x = 2 . 05 . 15 . 3 .5 Sayan Mukherjee Joint distributions and the central limit theorem
Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example p ( x , y ) y = 2 y = 3 y = 4 x = 1 . 2 . 1 . 2 x = 2 . 05 . 15 . 3 p Y ( y ) . 25 . 25 . 5 Sayan Mukherjee Joint distributions and the central limit theorem

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Joint distributions Definition Let X , Y be two continuous random variables. The joint pdf p ( x , y ) as the function that for any set A IP [( x , y ) A ] = ZZ A p ( x , y ) dx dy .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}