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Unformatted text preview: M4056 Lecture Notes. September 10, 2010 Transformations
These notes are intended to clarify a point in the last lecture.
Suppose we have two copies of Rn , one with coordinates x1 , . . . , xn and one with coordinates u1 , . . . , un . Suppose also that we have a pair of diﬀerentiable functions g and h that
establish a bijection between the two copies of Rn , with
u = g (x) and x = h(u).
This means that we can write each xi as a function of the uj s and each uj as a function
of the xi s.
Example. u T =A·x T for some invertible matrix A (so x T T = A−1 · u ). Now suppose X is a random variable with pdf fX (x). Then U := g (X ) is also a random
variable, and it has a pdf fU (u). What is the relationship between the two pdf s?
If E is a subset of Rn , then the change of variables theorem says: h(W ) fX (x)dx = W fX (h(u))|h′ (u)|du. Now,
h(W ) fX (x)dx = P (X ∈ h(W )) = P (g (X ) ∈ gh(W )) = P (U ∈ W ) Thus
P (U ∈ W ) =
W fX (h(u))|h′ (u)|du. Since this is true for all W ,
fU (u) = fX (h(u))|h′ (u)|.
See (4.3.2) on page 158 and the discussion around it. See also Example 4.6.13. ...
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