2 we suppose that there is a probability measure

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We suppose that there is a probability measure defined over R 2 such that, for any A ⊂ R 2 , P ( A ) is the probability that ( x, y ) falls in A . Thus, for example, if A = { a < x b, a < x b } , which is a rectangle in the plane, then P ( A ) = d y = c b x = a f ( x, y ) dx dy. This is a double integral, which is performed in respect of the two variables in succession and in either order. Usually, the braces are omitted, which is allowable if care is taken to ensure the correct correspondence between the integral signs and the differentials. Example. Let ( x, y ) be a random vector with a p.d.f of f ( x, y ) = 1 8 (6 x y ); 0 x 2; 2 y 4 . It needs to be confirmed that this does integrate to unity over the specified range of ( x, y ). There is 1 8 2 x =0 4 y =2 (6 x y ) dydx = 1 8 2 x =0 6 y xy y 2 2 4 2 dx = 1 8 2 x =0 (6 2 x ) dx = 1 8 6 x x 2 2 0 = 8 8 = 1 . Moments of a bivariate distribution. Let ( x, y ) have the p.d.f. f ( x, y ). Then, the expected value of x is defined by E ( x ) = x y xf ( x, y ) dydx = x xf ( x ) dx, if x is continuous, and by E ( x ) = x y xf ( x, y ) = x xf ( x ) , if x is discrete. Joint moments of x and y can also be defined. For example, there is x y ( x a ) r ( y b ) s f ( x, y ) dydx, where r, s are integers and a, b are fixed data. The most important joint moment for present purposes is the covariance of x and y , defined by C ( x, y ) = x y { x E ( x ) }{ y E ( y ) } f ( x, y ) dydx. If x, y are statistically independent, such that f ( x, y ) = f ( x ) f ( y ) , then their joint moments can be expressed as the products of their separate moments. 3
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Thus, for example, if x, y are independent then E { [ x E ( x )] 2 [ y E ( y )] 2 } = x y [ x E ( x )] 2 [ y E ( y )] 2 f ( x ) f ( y ) dydx = x [ x E ( x )] 2 f ( x ) dx y [ y E ( y )] 2 f ( y ) dy = V ( x ) V ( y ) . The case of the covariance of x, y , when these are independent, is of prime impor- tance: C ( x, y ) = E { [ x E ( x )][ y E ( y )] } = x [ x E ( x )] f ( x ) dx y [ y E ( y )] f ( y ) dy = { [ E ( x ) E ( x )][ E ( y ) E ( y )] } = 0 . These relationships are best expressed using the notation of the expectations operator. Thus C ( x, y ) = E { [ x E ( x )][ y E ( y )] } = E [ xy E ( x ) y xE ( y ) + E ( x ) E ( y )] = E ( xy ) E ( x ) E ( y ) E ( x ) E ( y ) + E ( x ) E ( y ) = E ( xy ) E ( x ) E ( y ) . Since E ( xy ) = E ( x ) E ( y ) if x, y are independent, it follows, in that case, that C ( x, y ) = 0. Observe also that C ( x, x ) = E { [ x E ( x )] 2 } = V ( x ). Now consider the variance of the sum x + y . This is V ( x + y ) = E [( x + y ) E ( x + y )] 2 = E [ { x E ( x ) } + { y E ( y ) } ] 2 = E [ x E ( x )] 2 + [ y E ( y )] 2 + 2[ x E ( x )][ y E ( y )] = V ( x ) + V ( y ) + 2 C ( x, y ) . If x, y are independent, then C ( x, y ) = 0 and V ( x + y ) = V ( x ) + V ( y ). It is important to note that If x, y are independent, then the covariance is C ( x, y ) = 0 . However, the condition C ( x, y ) = 0 does not, in general, imply that x, y are independent. A particular case in which C ( x, y ) = 0 does imply the independence of x, y is when both these variables are normally distributed. The correlation coefficient. To measure the relatedness of x and y , we use the correlation coefficient, defined by Corr( x, y ) = C ( x, y ) V ( x ) V ( y ) = E { [ x E ( x )][ y E ( y )] } ) E { [ x E ( x )] 2 } E { [ y E ( y )] 2 } .
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  • Spring '12
  • D.S.G.Pollock
  • Probability theory, probability density function, yj

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