## Binomial

Coin tosses, random walks, and spins in a magnetic field can all be described the same way. The number of heads, steps to the right, or spins aligned with the magnetic field is described by the *binomial distribution*. If the number of tosses, steps, or spins becomes large so the number can be considered to be continuous, the binomial distribution becomes the *Gaussian distribution*, which describes the fluctuations from mean values of macroscopic quantities in thermal equilibrium.

Program `Binomial` allows you to explore the properties of the binomial distribution.

**Problem: The binomial distribution**

- Calculate the probability
$P_N(n)$ for $p=q=1/2$ that $n$ spins are up out of a total of $N$ for
$N=4$ and $N=16$. Determine the mean values of
$n$ and $n^2$ using your values of $P_N(n)$. Although it is
better to first do the calculation of $P_N(n)$ by hand for small $N$,
you can use Program
`Binomial`to do the calculation. - Use Program
`Binomial`to plot $P_N(n)$ versus $n$ for larger values of $N$ and $p=q=1/2$. Determine the value of $n$ corresponding to the maximum of the probability and visually estimate the width for each value of $N$. What is your measure of the width? - One measure of the width is to use the two values of $n$ at which $P_N(n)$ is equal to half of its value at its maximum. What is the qualitative dependence of the width on $N$? Compare the relative heights of the maximum of $P_N$ for increasing values of $N$.
- Program
`Binomial`also plots $P_N(n)$ versus $n/\overline{n}$. Does the width of $P_N(n)$ appear to become larger or smaller as $N$ is increased? - Plot $\ln P_N(n)$ versus $n$ for $N=16$. Describe the qualitative dependence of $\ln P_N(n)$ on $n$. Can $\ln P_N(n)$ be fitted to a parabola of the form $A + B(n-\overline{n})^2$, where $A$ and $B$ are fit parameters?

## Resource

Problem 3.37 in*Statistical and Thermal Physics: With Computer Applications*, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).