02_Probability_part3

02_Probability_part3 - Ch1~6, p.25 Example 2.5 (sum of two...

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NTHU MATH 2820, 2008, Lecture Notes Ch1~6, p.25 Example 2.5 (sum of two discrete random variables, TBp. 96) (Exercise: difference of two random variables, Z=Y X ) X and Y are random variables with joint pmf p ( x,y ). Find the distribution of Z = X + Y . p Z ( z )= P ( Z = z )= P ( X + Y = z )= x = −∞ p ( x,z x ) When X , Y independent, p ( x, y )= p X ( x ) p Y ( y ) , p Z ( z )= x = −∞ p X ( x ) p Y ( z x ) convolution of p X and p Y X Y made by Shao-Wei Cheng (NTHU, Taiwan) Ch1~6, p.26 2. method of cumulative distribution function (a special case of method 1 ) Let Y be a function of the random variables X 1 ,X 2 ,...,X n . 1. Find the region Y y in the ( x 1 ,x 2 ,...,x n ) space. 2. Find F Y ( y )= P ( Y y )bysumm ingth ejo in tpm fo r integrating the joint pdf of X 1 ,X 2 ,...,X n over the region Y y . 3. (for continuous case) Find the pdf of Y by di f erentiating F Y ( y ), i.e., f Y ( y )= d dy F Y ( y ). Note. It can be generalized to multivariate Y =( Y 1 ,Y 2 ,...,Y m ).
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NTHU MATH 2820, 2008, Lecture Notes Ch1~6, p.27 Example 2.6 (square of a random variable, similar example see TBp. 61) For y 0 , { Y y } = { y X y } F Y ( y )= P ( Y y )= P ( y X y )= F X ( y ) F X ( y ) f Y ( y )= d dy F Y ( y )= d dy F X ( y ) d dy F X ( y ) = f X ( y ) 1 2 y f X ( y )( 1 2 y ) = 1 2 y ( f X ( y )+ f X ( y )) and f Y ( y )=0for y< 0. X is a random variables with pdf f X ( x )andcd f F X ( x ). Find the distributon of Y = X 2 . made by Shao-Wei Cheng (NTHU, Taiwan) Ch1~6, p.28 Example 2.7 (sum of two continuous random variables, TBp. 97) (Exercise: difference of two random variables, Z=Y X ) X and Y are random variables with joint pdf f ( x, y ). Find the distribution of Z = X + Y . Let R z be { ( x, y ): x + y z } .Then , F Z ( z )= P ( Z z )= P ( X + Y z )= R z f ( x, y ) dxdy = −∞ z x −∞ f ( x,y ) dydx = z −∞ −∞ f ( x,v x ) dxdv (set y = v x ) f Z ( z )= d dz F Z ( z )= −∞ f ( x, z x ) dx When X , Y independent, f ( x, y )= f X ( x ) f Y ( y ), f Z ( z )= −∞ f X ( x ) f Y ( z x ) dx convolution of f X and
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NTHU MATH 2820, 2008, Lecture Notes Ch1~6, p.29 Example 2.8 (quotient of two random variables, TBp. 98) (Exercise: product of two random variables, Z=XY ) X and Y are r.v. with joint pdf f ( x,y ). Find the distribution of Z = Y/X . Q z = { ( x,y ): y/x z } = { ( x, y ): x< 0 ,y zx } { ( x, y ): x> 0 ,y zx } F Z ( z )= Q z f ( x, y ) dxdy = 0 −∞ xz + 0 xz −∞ f ( x,y ) dydx = 0 −∞ −∞ z + 0 z −∞ xf ( x, xv ) dvdx (set y = xv ) = 0 −∞ z −∞ ( x ) f ( x, xv ) dvdx + 0 z −∞ xf ( x, xv ) dvdx = z −∞ −∞ | x | f ( x, xv ) dxdv f Z ( z )= d dz F Z ( z )= −∞ | x | f ( x, xz ) dx (= −∞ | x | f X ( x
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02_Probability_part3 - Ch1~6, p.25 Example 2.5 (sum of two...

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