Modeling Moments so Dear

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 dearas­sum­ing you ex­pe­ri­enced one im­me­di­ately be­fore this page load — this pro­vides a lower bound.

Table of Contents

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 is the mean, and 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 . 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 , the mean fol­lows the nor­mal dis­tri­b­u­tionpa­ra­me­ter­ized by the mean and the vari­ance (notably, vari­ance and not stan­dard de­vi­a­tion)

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 .

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?