chapter8 (1) - 8. Definition: .Then p(x,y)=P(X=x,Y=y *

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
          ________________________________________________________________________________ _   Chapter 8 Joint Distributions § 8.1 Bivariate Distributions  Joint Probability Functions • Definition: Let  X  and  Y  be two  discrete  random variables  defined on the  same sample space . Then p ( x , y ) =  P ( X = x , Y = y ) is called the  joint  probability function of  X  and  Y . * If the sets of all possible values of  X  and  Y  are called  A  and  respectively, then p X ( x ) =   Σ      p ( x , y       y  B p Y ( y ) =   Σ      p ( x , y       x  A •  Definitio n:  The above two functions are called the  marginal  probability functions of random variables  X  and  Y respectively. see Example 8.1       __________________________________________________________  © Shi-Chung Chang, Tzi-Dar Chiueh   ___         
Background image of page 1

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

View Full Document Right Arrow Icon
          ________________________________________________________________________________ _ * Example 8.2:  Roll a balanced die and let the outcome be  X Then toss a fair coin  X  times and let  Y  denote the number of  tails.    Solution p ( x , y ) =  P ( X = x , Y = y ) = (1/6)C( x , y )(0.5) x      p X ( x ) and  p Y ( y ) can be obtained from the table in text. Joint Probability Density Functions •  Definition : Two  continuous  random variables  X  and  defined on the same sample space have a continuous  joint  probability density function (jpdf)  if there exists a  nonnegative function of two variables  f ( x , y ) on  R x R , such that  for any region in the  xy -plane that can be formed from  rectangles by a countable number of set operations , P {( X , Y R } =  ∫∫ f X,Y ( x , y )   dx   dy . R The function is  f ( x , y ) called the joint probability density  function of  X  and  Y  . *   - - f X,Y ( x , y ) dx   dy  = 1 __________________________________________________________  © Shi-Chung Chang, Tzi-Dar Chiueh   ___         
Background image of page 2
          ________________________________________________________________________________ _ *   P ( X = a , Y = b ) =   a a b b f X,Y ( x , y dx   dy  = 0 *   P ( a  < X     b c  < Y     d ) =   a b c d f X,Y ( x , y )   dx   dy   *   f X ( x ) =  - f X,Y ( x , y dy   and   f Y ( y ) =  - f X,Y ( x , y dx •  Definition : The above two functions are called the  marginal  probability density functions of  X  and  Y , respectively.  
Background image of page 3

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

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 22

chapter8 (1) - 8. Definition: .Then p(x,y)=P(X=x,Y=y *

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online