Isye 2027

# If i j the situation is dierent the joint pmf of xi

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Unformatted text preview: t (W, Z ) is the image of (X, Y ) under the mapping from the u − v plane to the α − β plane deﬁned by α = min{u, v } and β = max{u, v }, shown in Figure 4.21. This mapping maps R2 into the set {(α, β ) : α ≤ β }, which is the set of points on or above the diagonal (i.e. the line α = β ) in the α − β plane. Since X and Y are jointly continuous, P {W = Z } = P {X = Y } = v ∞ −∞ v fX,Y (u, v )dudv = 0, so it does not matter how the joint density of (W, Z ) is deﬁned exactly on the diagonal; we will set it to zero there. Let U = {(α, β ) : α &lt; β }, which is the region that lies strictly above the diagonal. For any subset A ⊂ U, {(W, Z ) ∈ A} = {(X, Y ) ∈ A} ∪ {(Y, X ) ∈ A}, where the two sets in this union are disjoint. Therefore, using the fact fY,X (u, v ) = fX,Y (v, u), P {(W, Z ) ∈ A} = P {(X, Y ) ∈ A} + P {(Y, X ) ∈ A} = fX,Y (u, v ) + fY,X (u, v )dudv A = fX,Y (u, v ) + fX,Y (v, u)dudv A = fX,Y (α, β ) + fX,Y (β, α)dαdβ, A where in the last step we s...
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