x̄ - > Statistics, Applied mathematics and complex analysis

 

Statistics

Descriptive statistics

mean | {21.3, 38.4, 12.7, 41.6} = 28.5

ToeplitzMatrix[{21.3, 38.4, 12.7, 41.6}]

(21.3 | 38.4 | 12.7 | 41.6

38.4 | 21.3 | 38.4 | 12.7

12.7 | 38.4 | 21.3 | 38.4

41.6 | 12.7 | 38.4 | 21.3)

Dimensions - >4 (rows) × 4 (columns)

Properties -> symmetric, Toeplitz

Trace -> 85.2

Determinant - > -358459.

Inverse -> (0.0533794 | 0.0186717 | -0.0633426 | -0.00119043

0.0186717 | 0.059884 | -0.00489768 | -0.0633426

-0.0633426 | -0.00489768 | 0.059884 | 0.0186717

-0.00119043 | -0.0633426 | 0.0186717 | 0.0533794)

Characteristic polynormial - > λ^4 - 85.2 λ^3 - 3754.68 λ^2 + 81200.5 λ - 358459.

Elgen value 

λ_1 = 1/10 (613 + sqrt(261377))

λ_2 = 1/10 (-187 - sqrt(66305))

λ_3 = 1/10 (613 - sqrt(261377))

v_4 = (-1, 1/257 (-16 - sqrt(66305)), 1/257 (16 + sqrt(66305)), 1)

Diagonalization

M = S.J.S^(-1)

where

M = (21.3 | 38.4 | 12.7 | 41.6

38.4 | 21.3 | 38.4 | 12.7

12.7 | 38.4 | 21.3 | 38.4

41.6 | 12.7 | 38.4 | 21.3)

S = (-1 | -1 | 1 | 1

0.939679 | -1.06419 | -1.0318 | 0.969179

-0.939679 | 1.06419 | -1.0318 | 0.969179

1 | 1 | 1 | 1)

J = (-44.4498 | 0 | 0 | 0

0 | 7.04976 | 0 | 0

0 | 0 | 10.175 | 0

0 | 0 | 0 | 112.425)

S^(-1) = (-0.265534 | 0.249517 | -0.249517 | 0.265534

-0.234466 | -0.249517 | 0.249517 | 0.234466

0.242176 | -0.249878 | -0.249878 | 0.242176

0.257824 | 0.249878 | 0.249878 | 0.257824)

Condition number - 16.7347

Statistical inference

The sample size for estimating a binomial parameter

n = ((erf^(-1)(c))/(sqrt(2) M))^2 | 

n | sample size

M | margin of error

c | confidence level

the margin of error | 0.1

confidence level | 0.95

sample size | 96.04

T-interval for a population mean | 

sample mean | 4.15

sample standard deviation | 0.32

sample size | 100

confidence level | 0.95

95 % confidence interval. 4.087 to 4.213

x^_ ± (t_((1 - c)/2) s)/sqrt(n) = 4.15 ± 0.0634949 | 

n | sample size

s | sample standard deviation

x^_ | sample mean

c | confidence level

Regression analysis

fit | data | {{1.3, 2.2}, {2.1, 5.8}, {3.7, 10.2}, {4.2, 11.8}}

model | linear function

Least square best fit - > -1.52256 + 3.19383 x

AIC | BIC | R^2 | adjusted R^2

10.4041 | 8.56296 | 0.989771 | 0.984657

Random Variables

E(-7 + 3 X^4) where 

X distributed Poisson distribution | mean | μ = 7.3

16655.8

probability of A | 0.4

probability of B | 0.6

probability of A intersection B | 0.2

event | probability

P(A conditioned B) | 0.333333

P(B conditioned A) | 0.5

P(A intersection B) = P(A conditioned B) P(B)

P(A intersection B) = P(B conditioned A) P(A)

Applied Mathematics 

Game theory

tic-tac-toe (mathematical game) Assuming it is a mathematical instead of a board game

Two players alternately place pieces (typically X's for the first player and O's for the second) on a 3 × 3 board. The first player to get three matching symbols in a row (vertically, horizontally, or diagonally) is the winner. If all squares are occupied but neither player has three symbols in a row, the game is a tie.

naughts and crosses | three-in-a-row | ticktacktoe | wick wack woe

Payoff matrix

 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0

2 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0

3 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1

4 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1     

5 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 

6 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1

7 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1

8 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0

9 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0

(wherein the first matrix, each player would prefer winning over tying, tying over losing, and losing over crashing; in the second matrix, the benefit of winning is set to 1, the cost of losing is set to -1, and the cost of crashing is assumed to be -10)

fair | finite | futile | perfect information | sequential | two-player | zero-sum

chicken (mathematical game)

Two drivers travel in opposite directions on a collision course toward one another so that if at least one does not swerve, a collision will occur and both may be killed. However, if one driver swerves but the other does not, the swerving driver loses the game and is termed a "chicken, " i.e., coward.

hawk-dove | snowdrift

anticoordination | imperfect information | nonsequential | nonzero-sum | two-player

Fractals

Julia set -0.40+0.65i

Julia set | 

Re(c) | -0.4

Im(c) | 0.65

z_(n + 1) = z_n^2 + c | z_0 = z

(the Julia set is the boundary of the set of z element C for which the orbit of z_n is bounded)

Topological property

Julia set is totally disconnected

c = -0.4 + 0.65 i is not in the Mandelbrot set

packing and covering

estimate | number of baseballs | to fill | interior volume of a Boeing 747

container | Boeing 747

idealized shape | circular cylinder

interior volume | 1753 m^3

object | baseball

idealized shape | sphere

volume | 219 cm^3

packing density | (0.56 to 0.64)

Dynamical systems

logistic map | 

parameter r | 3.56994

initial condition x_0 | 0.1

logistic map | 

parameter r | 3.56994

initial condition x_0 | 0.1

Numerical analysis

solve x cos(x) = 0 using Newton's method to machine precision

x_(n + 1) = x_n - (x_n cos(x_n))/(cos(x_n) - x_n sin(x_n))

x = -1.187250704641182×10^-16

(using the starting point of x_0 = -0.017)

4th order iteration

x_(n + 1) = (x_n^2 (-6 x_n^4 sin^5(x_n) + 1/16 x_n^3 (74 cos(x_n) - 55 cos(3 x_n) + 29 cos(5 x_n)) - 12 sin(x_n) cos^4(x_n) + 14 x_n^2 (sin(3 x_n) - 2 sin(x_n)) cos^2(x_n) + 6 x_n sin^3(2 x_n) csc(x_n)))/(6 (cos(x_n) - x_n sin(x_n))^5)

3 steps to machine precision

solve y'(x) = -2 x^3 y(x)

y(1) = 5 using Euler method from x = 1 to 10

step | x | y | local error | global error

0 | 1 | 5 | 0 | 0

⋮ | ⋮ | ⋮ | ⋮ | ⋮

10 | 10 | 3.62614×10^21 | 0 | -3.62614×10^21

Optimization

Global maximum

max{x (1 - x) e^x} = (sqrt(5) - 2) e^(1/2 (sqrt(5) - 1)) at x = sqrt(5)/2 - 1/2

Complex Analysis

e^z

periodicity - > periodic in z with period 2 i π

1 + z + z^2/2 + z^3/6 + z^4/24 + O(z^5) 

(Taylor series) with Big- O - notation

Indefinite integral - >  integral e^z dz = e^z + constant

Definite integral - > integral_(-∞)^0 e^z dz = 1

lim_(z->-∞) e^z = 0 

Alternative represantation e^z = ζ^z for ζ = e

e^z = 1 + 2/(-1 + coth(z/2)) coth is the hyperbolic cotangent function

(1 + z)^a = ( integral_(-i ∞ + γ)^(i ∞ + γ) (Γ(s) Γ(-a - s))/z^s ds)/((2 π i) Γ(-a)) for (0<γ<-Re(a) and abs(arg(z))<π) 

γ - > 

Γ(s) gamma function

Re(a) Real part of Z

(arg(z) Complex argument of Z

i imaginary unit