求和化简疑问：为何∑_{y:y=x}P(x,y)=P(x,x)中求和符号被移除？

Great question—let's break this down step by step to clear up the confusion!

First, unpack the summation condition #

The notation $\sum_{y,:,y=x}$ means "sum over all values of $y$ that satisfy the condition $y=x$". For a fixed $x$, how many values of $y$ meet this condition? Exactly one: $y$ has to equal the specific $x$ we're working with right now.

Why the summation symbol gets removed #

A summation’s purpose is to add up all terms that match its condition. If there’s only one term to add, the summation is redundant—adding a single number is just the number itself.

In this case, the only term in the sum is when $y=x$, so we substitute $y=x$ into $P(x,y)$ to get $P(x,x)$. Since there’s nothing else to add, we can drop the summation symbol entirely; it doesn’t serve any mathematical purpose here.

Why your expected results don’t fit #

Let’s address the two outcomes you thought might happen:

• $\sum_{y,:,y=x} P(x,x)$: This is technically mathematically equivalent to $P(x,x)$, but it’s unnecessary to keep the summation. Since $P(x,x)$ doesn’t depend on $y$, summing it over the single valid $y$ value just gives you the same value. We simplify to $P(x,x)$ because it’s clearer and avoids redundant notation.
• $\sum_{all , x} P(x,x)$: This is a completely different operation. The original sum is for a fixed $x$, summing over $y$ (which only has one valid value). This version sums over all possible $x$ values, adding up every $P(x,x)$ in your domain—this isn’t what the original expression is asking for at all.

Quick recap #

The core idea is that the summation condition restricts $y$ to exactly one value. Summing a single term is identical to writing that term directly, so we remove the summation symbol for simplicity and clarity.

内容的提问来源于stack exchange，提问作者A_for_Abacus