Poisson Process

I know I've written in the past that a baby's poop can have serial correlation, but largely it follows a Poisson process

Early on Monday morning, my son “gave himself a baby shower”.

He had woken up at around 5am, crying incessantly, and refusing to either drink milk and going back to sleep. Based on the “ML algo” we’ve learnt over the last few months, we reasoned that he must have shat, and duly took him to the changing table.

He had duly “produced output”. And as soon as I opened him up, he continued to produce output. The thing with nappy pants (which we prefer for him, since my wife believes they work better for boys) is that once removed torn, they cannot be put back on. I immediately panicked, trying t put the top of the (now torn) diaper under his bum to catch his fresh output. It was only partly successful.

I was turning around to ask my wife for help, when he started peeing (a normal process once you’ve pooped, if you think of it).

When Billy was small, one of the great difficulties I had was in appreciating the new “angle of attack”. Having brought up a girl before him, I was used to pee going one way, and in a predictable way. What I had not reckoned then was that a little boy is capable of directing his output in all possible directions, including on himself.

Somehow I escaped getting his “baby shower”, but my wife wasn’t so lucky, having been showered at least twice when she changed his nappies. Even the lady who comes in to give his bath wasn’t spared of the shower. He even peed on his own face a couple of times (something our daughter clearly never faced (no pun intended) ).

In any case, when he was a month and a bit old, these showers stopped. He would dutifully do his business, and allow us to change him in peace. A couple of months later (now I’ve forgotten the timelines), he started giving me big smiles every time I took him to the changing station. He continues to do that even now, though a recent rearrangement in the bedroom has meant the changing table is right next to a power socket, which he tries to inspect!

When he was very young, I remember being paranoid about changing him real quick (when my daughter was very young, a friend had remarked that one should be able to change nappies in precisely 52 seconds - you can figure for yourself what else takes that long!). I’d keep a fresh diaper and wipe handy, to minimise the chances of getting showered while changing him.

You see, broadly speaking (though I myself have written to the contrary), a baby’s output can be modelled as a poisson process. The likelihood of him/her starting to pee/poop in a given short interval of time is constant, and independent. This means that the longer you keep him “open”, the greater the chances of your getting “showered”. In fact, up to a reasonable approximation, the chances of your getting showered increase linearly with the amount of time you keep him open.

What happens as a baby grows up is that he starts peeing / pooping less often. While the volumes might be higher when the events happen, the likelihood of the event happening in a given short interval of time becomes lower. This means that you can take two minutes to change, instead of one minute at a younger age, and still have a lower probability of getting showered (I haven’t precisely calculated the parameters of the Poisson process).

There was a period in April and May when I would think of this Poisson process every time I changed Billy. I would think to myself “the only reason I can be relaxed when changing him is that the likelihood he will start peeing at a given point in time is only getting lower as he grows”, and then go on to think “then again, some day I will get showered”.

Nevertheless I’ve continued to get complacent and become more relaxed while changing him.

And thinking about it, what happened on Monday morning was not an uncorrelated process. I basically caught him midway through the poop process, and so the chances that he would continue to poop as soon as I opened him up were higher than normal. Then I panicked and forgot that poop gets followed by pee, and let him give himself a baby shower.

The model was not wrong. Poisson process largely holds. It was that I failed to identify that this was a special case. In my defence, it was at ~5am.