Math/Stat
304: Worksheet: April 4th
Transformations: Bivariate Random Variables
Change of variables (Hogg-Craig)
Here we consider random variables of the continuous type; a similar method works for discrete random variables.
Only transformations that define a one-to-one (injective) transformation will be considered.
Let yI = uI(x1, x2) and y2 = u2(x1, x2) define a one-to-one transformation that maps a (two-dimensional) set A in the x1
x2 -plane onto a (two-dimensional) set B in the y1 y2 –plane.
If we express each of x1 and x2 in terms of y1 and y2 we can write x1 = w1 (yI, y2), x2 = w2 (yI, y2). The determinant
𝜕(𝑥1 ,𝑥2 )
of order 2, is called the Jacobian of the transformation and will be denoted by the symbol J. It will be
𝜕(𝑦1 ,𝑦2 )
assumed that these first-order partial derivatives are continuous and that the Jacobian J is not identically equal to 0.
Example:
Hint: Let Y2 = X2.
Then find the marginal pdfs of Y1 and Y2.
Example: