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From Jaynes' probability theory: the logic of science, I found this:

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$p$ here is the joint probability distribution of $x,$ and $y$. I'm assuming the $\times$ denotes the cartesian product, but I don't really understand what this equation means, nor why it captures the assumption that $x$ and $y$ are independent.

Why does this equation capture the assumption that $x$ and $y$ are independent?

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    $\begingroup$ I'd say $\times$ is multiplication of measures here. I wouldn't use the same $f$ for $x$ and $y$, though. $\endgroup$ – Hagen von Eitzen Jul 24 '17 at 6:08
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$\times$ means the regular multiplication between two real numbers, not the cartesian product.

The definition of independent is

$$P(X=x, Y=y) = P(X=x)P(Y=y)$$

You might want to understand the equation as

$$\int \int \rho(x,y) dxdy = \int f_X(x) dx \int f_Y(y) dy$$

Credit: Hagen for pointing out we should not use the same $f$ for both variables.

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I think it means that the probability density function $\rho(x, y)$ is given by $$\rho(x, y) = f(x) \cdot g(y).$$

This implies independence, because if $A \subseteq X$ and $B \subseteq Y$, then $$\begin{align*} \Pr(x \in A \wedge y \in B) & = \int \limits_{A \times B} \rho(x, y) \, \mbox{d} x \mbox{d} y = \int \limits_{A \times B} f(x) \cdot g(y) \, \mbox{d} x \mbox{d} y \\ & = \int \limits_{A} f(x) \mbox{d} x \cdot \int \limits_{B} g(y) \mbox{d} y \\ & = \Pr(x \in A) \cdot \Pr(y \in B) \end{align*}$$

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