How do I find the PDF from a multidimensional CDF with indicator functions?

Question

I have what I'm sure is a very stupid question. When I have a two-dimensional random variable $\tilde{X}=(X_1,X_2)$ with the cdf $F(x_1,x_2)=(kx_1^2I_{(0,1)}(x_1)+I_{[1,\infty)}(x_1))(kx_2^2I_{(0,1)}(x_2)+I_{[1,\infty)}(x_2))$, how can I calculate the pdf from that? I get that I have to differentiate somehow, but what do I do with the indicator functions?

Answer 1

When in doubt, deal with indicator functions (and other awkward creatures like absolute values) by breaking the problem into cases.

Here we have

$$F(x_1,x_2) = \begin{cases} (kx_1^2)(kx_2^2) & \text{if } x_1 \in (0,1) \text{ and } x_2 \in (0,1)\\ kx_1^2 & \text{if } x_1 \in (0,1) \text{ and } x_2 \gt 1 \\ kx_2^2 & \text{if } x_1 \gt 1 \text{ and } x_2 \in (0,1)\\ 1 & \text{if } x_1 \gt 1 \text{ and } x_2 \gt 1 \\ \end{cases}$$

Each of these is straightforward to differentiate.

Sometimes you'll be able to consolidate the cases afterwards. Here for instance, $\frac{\partial^2 F}{\partial x \partial y}$ is zero everywhere outside the unit square $0 \le x_1,x_2 \le 1$

Answer 2

Alternative way to describe the CDF without indicator functions

It is common to split up the CDF into cases like for the uniform distribution between $a$ and $b$ you have: $$F_X(x) = \begin{cases} 0 &:& x < a \\ \frac{x-a}{b-a} &:& a\leq x\leq b\\ 1 &:& x>b \end{cases}$$

and the density is the derivative $f_X(x) = {F_X}^\prime(x)$

$$f_X(x) = \begin{cases} 0 &:& x < a \\ \frac{1}{b-a} &:& a\leq x\leq b\\ 0 &:& x>b \end{cases}$$

You can get rid of the indicator in your function by describing it in the format above (only now you have to write the cases based on two variables together).

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The derivative of a 2D CDF to obtain a PDF

The CDF is differently defined in a 2D case. One way is like

$$F_{X,Y}(x,y) = P(X\leq x, Y \leq y) = \int_{-\infty}^x\int_{-\infty}^y f_{X,Y}(t,s) ds dt$$

So you need to compute the derivative for both variables

$$\frac{\partial^2}{\partial x\partial y} F_{X,Y}(x,y) = f_{X,Y}(x,y)$$