# Binomial Theorem

Galton Board: classic demonstration of binomial distribution, in which a series of equal outcomes (ball bouncing to left or right of a pin) produces a bell curve

### The Galton Board

Sir Francis Galton, 1822-1911, a British mathematician, invented a nifty toy, which demonstrates clearly that a series of pairs of outcomes (ball bouncing to left or right on pins) will produce the pattern known as the binomial distribution.

\$\$P(X=r)=(\table n;r)p^rq^{n-r}\$\$

where r = 0, 1, 2, ...., n, and \$(\table n;r)\$ ≡ nCr, \$P(X=r) ≡ P_r\$, and \$p\$ and \$q\$ are the respective probabilities of outcomes event \$p\$ and event \$q\$.

## Pascal's Triangle

```n = 0                1
n = 1             1     1
n = 2          1     2     1
n = 3        1    3     3     1
n = 4      1   4     6     4     1
n = 5   1    5    10    10    5     1
....
n = n   \$({\table n;0})\$   \$({\table n;1})\$   \$({\table n;2})\$   ...   \$({\table n;{n-2}})\$   \$({\table n;{n-1}})\$   \$({\table n;n})\$

```

These are the binomial coefficients of the expansion of any expression to the power of n.

## The General Binomial Theorem

\$\$(a + b)^n = ({\table n;0})a^n + ({\table n;1})a^{n-1}b + ({\table n;2})a^{n-2}b^2 + ... + ({\table n;{n-1}})ab^{n-1} + ({\table n;{n}})b^{n} \$\$

where \$({\table n;r})\$ is the binomial coefficient of \$a^{n-r}b^r\$, and r is any integer from 0 to max. n.

The general term in the binomial expansion of \$(a + b)^n\$ is: \$\$T_{r+1} = ({\table n;r})a^{n-r}b^r\$\$

where \$({\table n;r}) = {_n}C_r\$.

### Example

The fifth row of Pascal's triangle is: 1, 5, 10, 10, 5, 1.

The binomial expansion of \$(x+3/x)^5\$ is therefore:

\$1x^5 + 5(x^4)(3/x)^1 + 10(x^3)(3/x)^2 + 10(x^2)(3/x)^3 + 5(x^1)(3/x)^4 + 1(x^0)(3/x)^5\$

\$= x^5 + 15x^3 + 90x + {270}/x + {405}/{x^3} + 243/{x^5}\$

### Coefficients

The General Binomial Theorem may be used to quickly find the coefficient of a specific x term.

For example, the coefficient of \$x^3\$ in the expansion of \$(2x+4)^6\$:

\$(2x+4)^6\$: \$a= 2x\$, \$b=4\$, \$n=6\$

\$T_{r+1} = ({\table n;r})a^{n-r}b^r\$

\$n-r=3\$, so \$r=n-3=6-3=3\$

\$T_{4} = ({\table 6;3})(2x)^{3}4^3 = {6!}/{3!3!}2^3x^34^3 = (20)⋅8⋅64x^3 = 10,240x^3\$

### Mode and Median of the Binomial Distribution

The probability distribution of the random variable \$X\$ (number of successful outcomes from \$n\$ Bernoulli trials) is:

\$\$P(X=r) = (\table n;r ) p^rq^{n-r}\$\$

\$X\$ follows a binomial distribution with parameters \$n\$ and \$p\$, where \$X∼B(n,p)\$. The third parameter is \$q=1-p\$.

The mode of \$X\$ is where the function has a maximum.

The median of \$X\$ = \$m = {x_1 + x_2}/2\$, where \$x_1\$ is the maximum value for which \$F(x_1) ≤ 1/2\$ and \$x_2\$ the minimum value for which \$F(x_2) ≥ 1/2\$.

If \$X ∼ B(n, p)\$ then \$E(X) = μ = np\$ and Var(X) = \$σ^2 = npq\$, where \$q=1-p\$.

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