Example 486 suppose x1 xn are independent and

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Unformatted text preview: imply changed the variables of integration. Consequently, the joint pdf of fW,Z is given by fX,Y (α, β ) + fX,Y (β, α) α < β fW,Z (α, β ) = 0 α ≥ β. There are two terms on the right hand side because for each (α, β ) with α < β there are two points in the (u, v ) plane that map into that point: (α, β ) and (β, α). No Jacobian factors such as those in Section 4.7.2 appear for this example because the mappings (u, v ) → (u, v ) and (u, v ) → (v, u) both have Jacobians with determinant equal to one. Geometrically, to get fW,Z from fX,Y imagine spreading the probability mass for (X, Y ) on the plane, and then folding the plane at the diangonal by swinging the part of the plane below the diagonal to above the diagonal, and then adding together the two masses above the diagonal. This interpretation is similar to the one found for the pdf of |X |, in Example 3.8.8. 4.8. MOMENTS OF JOINTLY DISTRIBUTED RANDOM VARIABLES 4.8 151 Moments of jointly distributed random variables The first and second moments, or equivale...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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