Random Variable Transformation
Table of contents
Defining Random Variable Transformation
Let $X$ be a random variable.
Let $Y = r(X)$ be a function of $X$ (e.g. $Y = X^2$).
Then $Y$ is a random variable transformation of $X$.
Discrete Random Variable Transformation
Let $X$ be a discrete random variable with probability mass function $p_X(x)$.
Then the mass function of $Y = r(X)$ is:
$$ p_Y(y) = P(Y = y) = P(r(X) = y) = P(X \in r^{-1}(y)) = \sum_{x \in r^{-1}(y)} p_X(x) $$
So it really boils down to solving for $r^{-1}(y)$.
Continuous Random Variable Transformation
For continuous random variables, in order to find the probability density function (PDF) $f_Y(y)$, we first solve for the cumulative distribution function (CDF) $F_Y(y)$, and then take the derivative.
To do that, we first find the set:
$$ A_y = \{ x \mid r(x) \leq y \} $$
Then find the CDF:
$$ F_Y(y) = P(Y \leq y) = P(r(X) \leq y) = P(X \in A_y) = \int_{A_y} f_X(x) dx $$
Then take the derivative:
$$ f_Y(y) = \frac{d}{dy} F_Y(y) $$
Multivariate Transformation
Examples of multivariate transformation would be:
- $Z = X + Y$
- $Z = \min(X, Y)$
- etc.
The general idea is the same as the univariate case.
For $Z = r(X, Y)$, we first find the set:
$$ A_z = \{ (x, y) \mid r(x, y) \leq z \} $$
Then find the CDF:
$$ F_Z(z) = P(Z \leq z) = P(r(X, Y) \leq z) = P((X, Y) \in A_z)= \int \int_{A_z} f_{X, Y}(x, y) dx dy $$
Then take the derivative:
$$ f_Z(z) = \frac{d}{dz} F_Z(z) $$
Expectation of Transformed Random Variable
The Law of the Unconscious Statistician (LOTUS)
Univariate Case
When $Y = r(X)$:
$$ \E(Y) = \E(r(X)) = \int r(x) p_X(x) $$
Multivariate Case
When $Z = r(X, Y)$:
$$ \E(Z) = \E(r(X, Y)) = \int \int r(x, y) dF_{X, Y}(x, y) $$