In the time that you’ve started reading this, there’s a chance that you’ve experienced, as Jonathan Larson put it, a moment so dear1.

# How do you measure a year?

RENT is a Tony Award and Pulitzer Prize winning musical describing the bohemian lifestyle of a group of young artists in Lower Manhattan. It remains one of the longest running shows on Broadway.

Even if you haven’t heard of the musical, you’ve probably heard one of its most famous numbers: Seasons of Love, which has become a popular hit in its own right. Here is the iconic first verse:

Five hundred twenty-five thousand six hundred minutes
Five hundred twenty-five thousand moments so dear
Five hundred twenty-five thousand six hundred minutes
How do you measure, measure a year?

I’m less interested in how to measure a year, and more interested in getting my fair share of moments so dear.

# Send in the Clowns Stats

We know from the lyrics that there are 525,000 moments so dear in one year. There are 525,600 minutes. That means there is, on average, one moment so dear (MsD) approximately every 1.001 minutes.

It makes sense to model the distribution of our MsDs (MssD?) as exponential. Exponential distributions are frequently used to model “waiting time” problems, because they have a couple of properties, which are reflected in our scenario.

1. The random variable (in our case, the amount of time until a moment so dear, since the last moment so dear) is always positive. This makes sense, since we shouldn’t ever say something like “the next moment so dear is -3 minutes from the previous moment so dear”
2. The distribution is memoryless. That means it doesn’t matter if we have been waiting for 3 minutes or 3 hours, the probability of a moment so dear occurring in the next minute will always be the same. As the cast sings in Finale B, there is “no day but today”.

With those conditions, I claim our moments so dear can be modeled as:

where $\beta$ is the mean, and $M$ is the random variable for minutes until the next MsD.

At this point, you may be asking yourself, what are the implications?

The implications, my friend, are many.

# Calculating Probabilities

First, this means we can make claims about, say, the probability of experiencing a moment so dear.

In this figure, the black line represents the probability density function of our random variable $X$. The red line, which is more interesting for us, represents the cumulative density function. This value is the same as the blue area under the curve. You can interpret it as y is the probability that we experience a moment so dear in the first x minutes.

So if we want to know the probability of experiencing a moment so dear in the five minutes following our previous MsD, we can plug in minutes and solve for y = .

So there’s a 99% chance that a moment so dear will occur within 5 minutes of the previous one.

# Confidence Intervals

We might also be curious how confident we are in this model. After all, who’s to say there are always 525,000 moments so dear in a year? We should naturally expect there to be some variance – we might get some more dear moments in one year, in a week, indeed in a minute, than the next!

Assuming we’re right to use the exponential distribution, how good is our guess of the parameters (in this case, the average waiting time)? Since we’re looking at means, we can use the Central Limit Theorem to give a rough confidence interval. We know, for large $n$, the mean follows the normal distribution2

We can use this mean and standard deviation to calculate a 95% confidence interval: we can state with 95% confidence that the true mean of our exponential distribution lies in the interval $[ 0.9984, 1.004 ]$.

# Life Lessons

If you are like me, you may have been surprised by just how frequently these moments so dear arise. After all, since you’ve loaded this page, there’s a chance that you’ve experienced a moment so dear.

I certainly didn’t think that was the case, but maybe there’s a lesson to be learned here. Maybe I do experience a moment that I should hold dear every minute or so. Maybe we should cherish more of the little things and embrace the delight that comes with our normal everyday life.

After all, the numbers never lie, right?