Money

Imagine a room full of \(N=100\) people with \(D=100\) dollars each. With every tick of the clock, every person with money gives a dollar to one randomly chosen other person. After some time progresses (number of runs equals to 50000), how will the money be distributed?

The Normal Curve in the histogram is based on the Normal Aproximation for the Binomial distribution.

\[X_i = \begin{cases} 1, & \mbox{if person $i$ gave me 1 dollar} \\ 0, & \mbox{otherwise } \end{cases}\]

\[p=P(X_i=1)=\frac{1}{N-1}\]

So that \(X_i\sim Bernoulli\left(p=\frac{1}{N-1}\right)\).

Let \(Y_r=\sum_{i=1}^{N-1}X_i\) be the total dollars I received at run \(r\).

\[Y_r\sim Binomial\left(n=N-1\,,p=\frac{1}{N-1}\right)\]

Let \(Z=\sum_{r=1}^R Y_r\) be the total dollars I received at the end of \(R\) runs.

\[Z\sim Binomial\left(n=(N-1)\times R\,,p=\frac{1}{N-1}\right)\]

We can use the Normal Distribution approximation to the Binomial:

\[Z \sim Normal\left(\mu=(N-1)\times R \times \frac{1}{N-1}, \sigma^2=(N-1)\times R \times \frac{1}{N-1} \times \frac{N-2}{N-1}\right)\]

\[Z \sim Normal\left(\mu= R, \sigma^2=R \times \frac{N-2}{N-1}\right)\]

Let \(W\) be the amount of dollars after the \(R\) runs.

\(W=Z-R+D\) since I gave away a dollar at each run and started with \(D=100\) dollars.

\[W \sim Normal\left(\mu= D, \sigma^2=R \times \frac{N-2}{N-1}\right)\]

The curve in the histogram is the distribution of \(W\), how much a person has at the end of \(R = 50000\) runs, given that she started with \(D=100\) dollars and was allowed to unlimited debt.