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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 ) =
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.
- Spring '08
- The Land