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An example of finding the joint probability distribution of four random variables x1, x2, x3, and x4, and then applying a multivariate transformation to obtain the joint probability distribution of the new random variables u1, u2, u3, and u4. The joint probability density function (pdf) of the original variables, as well as the steps to find the marginal and conditional pdfs, the expected value of the product of two variables, and the inverse transformation. The example uses the given joint pdf and applies the multivariate transformation to find the joint pdf of the new variables.
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Example 3. Let n = 4 and the joint density of (X 1 , X 2 , X 3 , X 4 ) is
f(X 1 ,X 2 ,X 3 ,X 4 )(x 1 , x 2 , x 3 , x 4 ) =^34 (x^21 +x^22 +x^23 +x^24 ), if 0 < xi < 1 , i = 1, 2 , 3 , 4;
and = 0 otherwise.
(i) Show that this is a valid pdf.
(ii) Compute P (X 1 < 12 , X 2 < 34 , X 4 > 12 )
(iii) Obtain the marginal pdf of (X 1 , X 2 ).
(iv) Find the conditional pdf of (X 3 , X 4 ) given X 1 = 13 and X 2 = 23.
(v) Compute E(X 1 X 2 ).
Multivariate Transformation Let (X 1 , · · · , Xn) be a random vector with pdf fX 1 ,···,Xn (x 1 , · · · , xn). Let A = {x : fX(x) > 0 }. A new random vector (U 1 , · · · , Un) is defined by
U 1 = g 1 (X 1 , · · · , Xn), U 2 = g 2 (X 1 , · · · , Xn), · · · · · · Un = gn(X 1 , · · · , Xn).
The transformation is one-to-one from A onto B. The inverse of gi’s are
X 1 = h 1 (U 1 , · · · , Un), X 2 = h 2 (U 1 , · · · , Un), · · · · · · Xn = hn(U 1 , · · · , Un).
Let J be the Jacobian from the inverse. The joint pdf of U 1 , · · · , Un is then
fU 1 ,···,Un (u 1 , · · · , un) = fX 1 ,···,Xn (h 1 (u 1 , · · · , un), · · · , hn(u 1 , · · · , un))|J|.
Example: Let (X 1 , X 2 , X 3 , X 4 ) have the joint pdf
fX 1 ,X 2 ,X 3 ,X 4 (x 1 , x 2 , x 3 , x 4 ) = 24e−x^1 −x^2 −x^3 −x^4 , 0 < x 1 < x 2 < x 3 < x 4 < ∞.
Consider the transformation
U 1 = X 1 , U 2 = X 2 − X 1 , U 3 = X 3 − X 2 , U 4 = X 4 − X 3.
