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 $q$.

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.

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

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

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

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

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

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 $r=n-3=6-3=3$

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

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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Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on ScienceLibrary.info.

1826 - 1866

Bernhard Riemann was a German mathematician whose revolutionary ideas of multi-dimensional space challenged Euclidean geometry perspectives, and ultimately led mathematics to radically new approaches, and found applications in many fields, such as General Relativity.

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