Visualizing the beauty in physics and mathematics
A Galton board with 10 rows can be thought of as a binomial experiment, where a ball has 50% chance of moving to the left or right at each row. The number of steps the ball takes to the right follows a binomial distribution.
In a binomial distribution, the number of successes (steps to the right) in $n=10$ trials is represented by:
$𝑋 \approx \text{Binomial}(n=10, p=0.5) = \begin{pmatrix}n \ k\end{pmatrix}p^k(1 - p)^{n-k} = \begin{pmatrix}n \ k\end{pmatrix}p^n = \begin{pmatrix}10 \ k\end{pmatrix}0.5^{10}$
For higher and higher $n$, we gradually approach the normal distribution. This is also known as the central limit theorem
The probability density function that is known as the normal or Gaussian distribution is given by
$f(x) = \dfrac{1}{\sqrt{2\pi\sigma^2}}e^-\dfrac{(x-\mu)^2}{2\sigma^2}$
The parameter $\mu$ is the mean or expectation of the distribution, the parameter $\sigma$ is the variance.
For our binomial distribution, the mean $\mu$ is given by
$\mu=n \cdot p = 10 \cdot 0.5 = 5$
The variance $\sigma^2$ is given by
$\sigma^2=n \cdot p \cdot (1 - p) = 10 \cdot 0.5 \cdot 0.5 = 2.5 \Rightarrow \sigma = \sqrt{2.5}$
So we finally obtain:
$X = \text{Normal}(\mu=5, \sigma=\sqrt{2.5})$
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