If the random vector (X, Y, Z) has density f on R^3, then the random vector (X+Y/2, Y+Z/2, Z+X/2) has density given by

(B) g(u,v,w) = 4f(u-v+w, u+v-w, -u+v+w)

Q38Section I (Objective (Q1–Q80))38 of 80

Probability — density of a linear transformation via the Jacobian

If the random vector $(X, Y, Z)$ has density $f$ on $\mathbb{R}^3$ , then the random vector $\left(\dfrac{X+Y}{2}, \dfrac{Y+Z}{2}, \dfrac{Z+X}{2}\right)$ has density given by

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