Can I replace ##X_n = i## with ##A## to type less? Rules of math.

When working with random variables, it is tempting to make substitutions with placeholders, by writing writing ##A## instead of ##X_n=i##, because it greatly simplifies the look. It seems that if ##A## has all of the attributes of the equation ##X_n=I##, then such substitutions should be allowed because it is not logically inconsistent.

For example, I might replace
##(X_{n+1}=j )\text{ with } A##
##(X_{n}=i)\text{ with } B##
and
##(X_{n-1}=k)\text{ with } C.##
Then I could write ##P(A|B,C)=P(A|B)## because ##\{X_n\}## is a Markov chain.

But, if ##A, B##, and ##C## were arbitrary sets, then the above equation would not be true in general. So is this very very lazy and frowned upon mistake that only first year undergraduates make? It seems like my professor became stern because I did this on the last assignment, maybe she was angry that I was so lazy?

docnet said:

When working with random variables, it is tempting to make substitutions with placeholders, by writing writing ##A## instead of ##X_n=i##, because it greatly simplifies the look.

There's a saying: "Make things as simple as possible, but no simpler."
Replacing an equation that contains important information with a single letter seems to me to be oversimplification.

docnet said:

It seems that if ##A## has all of the attributes of the equation ##X_n=I##, then such substitutions should be allowed because it is not logically inconsistent.

For example, I might replace
##(X_{n+1}=j )\text{ with } A##
##(X_{n}=i)\text{ with } B##
and
##(X_{n-1}=k)\text{ with } C.##
Then I could write ##P(A|B,C)=P(A|B)## because ##\{X_n\}## is a Markov chain.

Doesn't seem like a good idea to me.

docnet said:

But, if ##A, B##, and ##C## were arbitrary sets, then the above equation would not be true in general. So is this very very lazy and frowned upon mistake that only first year undergraduates make? It seems like my professor became stern because I did this on the last assignment, maybe she was angry that I was so lazy?

Or she might have been peeved that your simplification was one that discarded important information.

Replacing a given, specific, equation with a symbol is usually a bad idea. You can represent general concepts with symbols, especially in logic. You can sometimes represent sets and events in probability with a letter to represent the set. It's usually better to show specific equations exactly. If your motivation to do it is out of laziness, then there is no excuse. In you example, because you do not want to type the equations, I need to look back at what A, B, and C are to determine if your probability statement makes sense. If the definitions were not right above, it would be very annoying.

If your symbols indicate the meaning of the equation/statement that would be much different from your A, B, C example, which conveys no meaning. I'm not sure if there is any real significance to the values of i, j, and k, in your example. If not, for a Markov process, you can say that ##S_k## indicates the state at step ##k## and ##P(S_{n+1} | S_n, S_{n-1}) = P(S_{n+1} | S_n)##.

So it is ill-advised to do. I will never do it again. It is so strange that I did that in the first place. I think I was feeling physical pain from having to sit on the keyboard and type, but I could not justify it afterwards because there is really no reason to take such a bad looking shortcut.

Out of curiosity, Is there a situation where something like it would make sense, representing logic statements with a single symbol?

docnet said:

Out of curiosity, Is there a situation where something like it would make sense, representing logic statements with a single symbol?

Yes. It is often done in logic. And in general math, equations are often given numbers so that they can be referred to later in the text. But notice that you are not using A, B, and C, to represent equations. They are representing events, described using equations.