There are a lot of interesting events that occur at time limits. For instance, the arriavals of busses at a bus stop, accidents on a highway, goals scored in a soccer (football) game. The processes that model such time limit events are called “point processes”. A vital consideration in these processes is how long it takes from one event to the subsequent. For instance, given you only missed a bus, how long will you’ve to attend for the subsequent bus? This time is a random variable and the alternative of the random variable specifies the purpose process. One alternative for this random variable is something that isn’t very random (deterministic numbers are only a special case of random ones). Busses arriving on a punctual schedule, every 10 minutes for instance. This might sound like the best possible point process, but there’s something even simpler. And it arises when the times between events follow an exponential distribution (the method is known as the Poisson process). It’s called the exponential distribution for good reason. It’s tied to Euler’s number, e and compound interest. In this text, we’ll see the connection.
Say you deposit 1$ within the bank. The rate of interest is x per yr. At the top of the yr, your balance shall be (1+x). To get more cash, you ask the bank to pay you the interest monthly as an alternative of yearly. Because the rate is x per yr, the interest you’ll earn in a month shall be x/12. And also you immediately re-invest the interest. So for the second month, your investment becomes (1+x/12) and this in-turn grows by an element of (1+x/12) meaning the quantity after 2 months is (1+x/12)². Repeating this for 12 months, your balance at the top of the yr shall be (1+x/12)¹². Using the Binomial theorem, this recent balance at the top of the yr is:
We will see that that is greater than the (1+x) we ended up with before. This is sensible since we were getting interest throughout the months and the interest was re-invested and earning further interest on top. But why stop at 12 intervals? You ought to compound as often as possible. Every millisecond if the bank will allow it. As a substitute of 12 intervals, we generalize to n intervals and make n really large. After every interval, our balance grows further by (1+x/n)^n. And at the top of the yr the quantity we’ll have,
Expanding this out with the binomial theorem,
As n becomes larger, the n-1, n-2, etc. are practically similar to n. So, all those terms involving n cancel out between the numerators and denominators (since we now have n→∞) and we’re left with:
If we differentiate B(x) with respect to x, we get B(x) back. If we plug in x=1, we get a really special number. Are you able to guess? It’s readily apparent from the primary two terms that this number is larger than 2.
We’ve just re-discovered the famous Eulers number, e=2.71828... And it seems, B(x)=e^x. This wasn’t immediately obvious to me, but we are able to see this by going back to equation (1).
Now we have e within the second equation, but not in the primary one. The x/n term contained in the bracket is type of getting in the best way of that. to wash it up, let’s change up the variables by defining:
This may make equation (1):
Note that taking the x outside the limit like we did above is allowed for continuous functions.
In order that was compound interest and the motivation for the number e. How does all this relate to point processes and the exponential distribution? The exponential distribution works in continuous time and models the time until some event (like a automobile accident).
The very best approach to understand it’s to consider the limit of tossing coins.
The claim to fame of the exponential distribution is the incontrovertible fact that it’s memory less. The truth is, it’s the only continuous distribution that’s memory less. When you’re waiting for a bus whose time until arrival is exponentially distributed, then it doesn’t matter how long you’ve waited already.
The distribution of the extra time you’ve to attend is strictly the identical weather you only arrived or have been waiting for ten hours. This property makes the exponential distribution very easy to work with.
Its easier to know this property after we make things discrete. As a substitute of waiting in continuous time for a bus to reach, imagine tossing a coin every minute and waiting to see a heads. The variety of tails we’ll see before we see the primary heads is a discrete random variable since it could possibly take only non negative integer values (unlike the bus arrival time which will be any real number like 3.4 minutes). This discrete distribution is known as the Geometric distribution.
An actual world scenario where the Geometric distribution applies perfectly is a slot machine in a casino which a gambler keeps playing until he hits a jackpot. Every spin of the machine is independent of the spins up to now. Which suggests that this Geometric distribution can also be memory less. When people think that a machine hasn’t yielded jackpot for a very long time and so one is “due”, they aren’t accounting for the memory-less nature of the method and falling prey to the “gamblers fallacy”.
We will model each spin of the machine by the toss of a coin. The coin has a probability p of heads. We start tossing this coin. What’s the probability that we haven’t seen a heads after k tosses? This simply implies that we’ve seen k consecutive tails. The probability of that is:
Now, we would like to maneuver to continuous time. So, we split a continuous timeline into discrete parts. The coin tosses occur at each of those discrete events. Each unit interval of t is split right into a large number, d of discrete parts.
And now we denote by T the time at which an interesting event (coin coming up heads) occurs. To get the distribution of T, we again goal its survival function, the probability that it is larger than some number, t. We all know that a complete of [t/d] tosses should have happened by this time (where [.] is the best integer function). For instance, if t=10 and every unit interval of time is split into 3 parts, then [10/3] = [3.33] = 3 tosses would have happened by then. To make this a very continuous time process, we’d like to make d so small that it vanishes. But as we make d small, we find yourself with an increasing variety of coin tosses. So, our p must also turn into small to compensate (otherwise, the events will turn into so frequent that any minuscule interval of time may have many events). So, the p and d variables must go to 0 concurrently. Using the equation above for the discrete case, the variety of tosses which have happened by time t and the incontrovertible fact that our p and d variables must go to 0 we get the survival function of T:
The second limit just becomes 1.
This limit is interesting only when p and d decrease to 0 together in a linear relationship with one another. Because each are going to zero together, the road has to have an intercept of 0. Let’s say the road is:
That is the step most individuals have trouble with. Why hastily this equation? Where did it come from? If we ask what the double limit in equation (3) equals, the reply goes to be that “it depends”. Depends upon the connection between p and d. For one thing, we all know that p and d are approaching 0 together. So the connection between them must go through (0,0). Next, we’d like to choose the functional form between them. And it’s as much as us what to choose. But when we pick anything but a linear relationship, we get a trivial answer (like 0 or 1 for any t) and don’t get an interesting continuous distribution.
With the linear relationship above, equation (3) becomes:
This limit is interesting only when p and d decrease to 0 together in a linear relationship with one another. Because each are going to zero together, the road has to have an intercept of 0. Let’s say the road is:
Equation (3) above becomes:
We want one other substitution to make this align with equation (1):
Which is the survival function of the exponential distrubution. Now we have gone here from the Geometric distribution, taken the limit and derived the exponential distribution, all while using results derived from compound interest.