The next approximation is for the probability that one proc happens at a particular time t. This event is mirrored by a mathematical process which is known as a probability density function.In probability theory, the chance of anything occurring or the addition of the probability of a number of different states always sums up to 1. For example, you pick a ball from a bag with a blue, red and yellow ball. The chance that you pick a ball (disregarding the color) is 1 because we’ve already established that you’re picking a ball. The chance you pick a ball of a particular color is 1 out of 3 because there are 3 different colored balls.
The integral in the equation shown is a fancy way of summing up all possibilities to 1 as I’ve stated. P(t) is a probability distribution function which is simply what the probability is over a range of t values. At one single t, there will be one unique probability (chance to proc).
Recall the equations from last time:
The paper has reasoned the equation for a proc at time t, by multiplying the chance for it not to proc between time [0,t) by the chance for it to occur between [t, t+dt]. ‘dt’ or adding a ‘d’ in front of a variable such as t, is a fancy way to say instantly or some like to say an infinitesimally short period of time. It’s used to describe an event occurring at time t. It’s a little more complicated than this but this is the essential gist of it.
From the equations above, the chance of a proc occurring at time t-t+dt means it could not have procced before time t, and must proc in that range. Thus we multiply the two equations from above:
By integrating this equation from 0 to infinity with respect to t, we get the probability that a proc occurs at any time t. Note in the above equation, that the interval t-t+dt is symbolized by a triangle which is what mathematicians like to use to describe change. Here it represents a change in time. With some math, this equation becomes:
The paper states this is in the form of a basic exponential distribution. Basically, there are some distributions or functions which commonly occur so they are recognizable. This equation, when plotted out, symbolizes one of these known as the exponential distribution (http://en.wikipedia.org/wiki/Exponential_distribution). The paper also mentions that the inter-proc time is distributed according to P(t) which is essentially saying in real English what we’ve just calculated. Between any two procs, there is a time where it does not occur, here we’ve denoted it as the period [0,t) and a time which it does occur [t,t+dt]. We are calculating the probability of a proc at this time [t,t+dt], therefore over the scale of times, which will be see from plotting the above function P(t) over a range of t, the probability to proc at certain times. From this plot, we might see a significant peak. This is what the average inter-proc time will be.
The next approximation is the general case of n events occurring in a time interval.
The author mentions it can happen in two cases.
1. All n events occur before time t, no events occur in time t – t+dt, approximated by the equation:
2. n-1 events happen before time t, 1 event in time t—t+dt
Note that the variables are used here so that the same two cases can occur for any time t. Combining these equations we arrive at the final equation:
We also know, from properties of differential functions that this holds true:
As I mentioned before, approximations like this are ways of calculating the value of a function like this without using difficult or complicated mathematics.
Solving this equation for Pn(t) yields:
(! is the mathematical symbol for factorial. 3! = 3 * 2 *1)
This is as mentioned, the standard form of another common distribution called the Poisson distribution, poisson being the name of a famous mathematician. The author mentions something important here. This chance of n procs of interval [0, t] is independent of your attack speed.
I think I’ll stop at this point. This section had a lot mathematical terms in it but I hope I’ve made it a little bit clearer to you. Similar to last time, I want to remind everyone that these equations are actually usable in real life. Often mathematicians forget about the purpose of their approximations and equations. Suppose you wanted to see how many procs you would get in an 8 minute fight. Given the constants for the trinket and your haste, you can plot the function out above and see what the distribution looks like. Then you can say with reasonable certainty, the number of procs you will get falls somewhere along the distribution you plotted. This is what makes math cool.
Look out for the last part of the explanation coming soon 🙂