# The definition of marginal distribution

For a product space $\Omega = \Omega_1 \times ... \times \Omega_n$ and its distribution (a probability function) $P$,

the $i$th marginal distribution is:

$$P_i(\omega_i)=\sum_{t\in \Omega: t_i=\omega_i} P(t)$$ where $\omega_i \in \Omega_i$.

What does the condition ${t\in \Omega: t_i=\omega_i}$ mean?

Here it's claimed that the marginal distributions appear in the "margins" of the following table:

I don't see the connection to the equation. What is the $i$th margin distribution in that table?

$t$ is an $n$-tuple $(t_1,t_2,\ldots,t_n)$ whose $i$-th component is a member of $\Omega_i$. The sum is over all $t$ such that the $i$-th component is a fixed element $\omega$ of $\Omega_i$, not $\omega_i$ as it is written. The notation leaves much to be desired.
• Dilip, "$\omega_i$" appears correctly in the notation, which has the desirable property of being explicit about which margin is involved. Unfortunately it does overburden the poor subscript "$i$", which it uses both for naming various objects ($\Omega_i$, $P_i$, and $\omega_i$) and to represent the image of the canonical projection $\pi_i:\Omega\to\Omega_i$ via the shorthand $t_i = \pi_i(t)$. Presumably this is because the author thinks of each $t\in\Omega$ implicitly as a tuple $(t_1,\ldots,t_i,\ldots,t_n)$. – whuber Nov 8 '15 at 14:59