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Binomial Theorem

Galton Board
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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