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 $q=1-p$.

Content © Renewable-Media.com. All rights reserved. Created : January 1, 2015

The most recent article is:

View this item in the topic:

and many more articles in the subject:

Physics is the science of the very small and the very large. Learn about Isaac Newton, who gave us the laws of motion and optics, and Albert Einstein, who explained the relativity of all things, as well as catch up on all the latest news about Physics, on ScienceLibrary.info.

1723 - 1790

Adam Smith was a Scottish economist, or indeed 'the Father of Modern Economy'. He penned the Wealth of Nations, published in 1776, a not insignificant year for revolutionary ideas...

Website © renewable-media.com | Designed by: Andrew Bone