Change of Random Variables
Sources:
- Jeseph K. Blitzstein & Jessica Hwang. (2019). Conditional propability. Introduction to Probability (2nd ed., pp. 369-375). CRC Press.
Change of Random Variables
Notation
Symbol | Type | Description |
---|---|---|
Random variable | Original continuous random variable | |
Function | Probability density function (PDF) of |
|
Random variable | Transformed continuous random variable | |
Function | Probability density function (PDF) of |
|
Function | Transform function, assumed to be differentiable and strictly monotonic | |
Function | Inverse of the transformation function |
|
Vector | Original random vector |
|
Function | Joint PDF of |
|
Vector | Transformed random vector |
|
Function | Joint PDF of |
|
Matrix | Jacobian matrix of partial derivatives of |
|
Scalar | Determinant of the Jacobian matrix | |
Sets | Open subsets of |
Abbreviations
Abbreviation | Description |
---|---|
Probability Density Function | |
CDF | Cumulative Distribution Function |
r.v. | Random variable |
Jacobian | Matrix of partial derivatives |
Change of variables in one dimension
Theorem: Change of variables in one dimension Let
where
Proof:
Let
Change of variables in multiple dimensions
Theorem: Let
Define
Assuming all partial derivatives
If
This holds for all