# Deriving the transformation function of a random variable from the original and the final distributions

Consider a random variable $X$ and consider that this variable can be either real or integral (so I would like to cover both cases: continuos and discrete random variables). Consider to transform this variable using a certain function $g(\cdot)$:

$$Y = g(X)$$

Consider to know both $f_X(x)$ and $f_Y(y)$: the probability mass functions of the original r.v. and the final (transformed) r.v.

In better words, I am asking you to consider a random variable and to apply a transformation. Generally we know the transformation, but in our case we know that we want to transform the original r.v. in order to get a final r.v. that we already know!

Is it possible to calculate $g(\cdot)$ from these conditions?

### PS

Of course I know that there are closed expressions to express the final pdf in terms of $g$ and $f_X(x)$. For a continuos r.v.:

• When $g$ in monotonic increasing: $f_Y(y) = f_X(g^{-1}(y)) \cdot \frac{\partial}{\partial y}g^{-1}(y)$.

• When $g$ in monotonic decreasing: $f_Y(y) = -f_X(g^{-1}(y)) \cdot \frac{\partial}{\partial y}g^{-1}(y)$.

For a discrete r.v.: $f_Y(y) = f_X(g^{-1}(y))$.

But I cannot get $g$ from these relations being this term inside a function composition.

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For continuous random variables, the easiest way to approach your problem is in two steps. First transform $X$ into a random variable $Z \sim U(0,1)$, and then transform $Z$ into $Y$. The transformations needed are $$Z = F_X(X), ~~ Y = F_{Y}^{-1}(Z)$$ where $F_X(\cdot)$ is the cumulative probability distribution function of $X$ and $F_Y(\cdot)$ is the cumulative probability distribution function of $Y$. If the inverse function $F_{Y}^{-1}$ is not uniquely defined, one needs to pay special attention.
If $X$ and $Y$ are discrete random variables, then it might be that there are no transformations that will map $X$ into $Y$. For example, $X$ is a Bernoulli random variable while $Y$ is a binomial random variable with parameters $(n,p)$ where $n > 1$.
Just a question, how can I make the transformation when X and Y are discrete? $F_Y^{-1}(y)$ cannot be done... – Andry Mar 19 '13 at 8:03