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Exponentials Radicals and Logs

幂（真数）为自变量，指数为因变量，底数为常量的函数

• Exponentials 指数, Radicals底数, and Logs对数
• standard arithmetic operations
• division, multiplication, addition, and subtraction.

Exponentials

• squaring a number; in other words, multipying a number by itself
• in other words, 5 x 5 x 5, or 5-cubed

 4（7）=16384

• 4 is the base, and 7 is the power or exponent in this expression
• Radicals (Roots)
  • ?2=9
  • ？ radical, or root
• square root
• cube root

Logarithms

 4?=16

Polynomials 多项式

• factorization: common factors （ greatest ）

Quadratic Equations

• palabola
• vertex

• Domain of a Function {x∈IR

  x≠0}

• Range of a Function {p(x)∈IR

  p(x)≥1}

• slope, limits

• curve, diffentible
  • Derivative function 导数 - max, min, or inflexion point - second-order Derivative

Multivariate Differentiation

find the analog of the slope for multi-dimensonal surfaces. We call this quantity the gradient.

Integration 积分

Integrals are the inverses of derivatives.

Vector 向量

• r, length or magnitude of vector
• Theta = tan-1(y/x), angle of the vector,
• addition, 3* multiplication
• multiplication
  • scalar: w = 2v
  • dot product (for scalar product):

  v*s =(v1⋅s1)+(v2⋅s2)…+(vn⋅sn) can use it to calculate the cosine of the angle between two vectors. v *s = |v| |s| cos(θ)

  • cross product(vector product))
  The result of this is a new vector that is at right angles to both the other vectors in 3D Euclidean space. p3 * q3 = r3:

    r1=p2q3−p3q2, 
    r2=p3q1−p1q3, 
    r3=p1q2−p2q1
  

Matrice

• negative
• transposition: row and column orientation 转至
• mulitplication identity matrics 对角线为1: D *I = M D(t)
• Division: reciprocal or inverse N(-1) 逆 A / B = A * B(-1) B * (B-1) = B(t) =I
• Transformations, Eigenvectors, and Eigenvalues
  Matrices and vectors are used together to manipulate spatial dimensions
  • Linear Transformations T(v)=Av calculate the dot product by applying the RC rule; T: Rm -> Rn

  • Transformations of Magnitude and Amplitude
    When you multiply a vector by a matrix, you transform it in at least one of the following two ways: - Scale the length (magnitude) of the matrix to make it longer or shorter - Change the direction (amplitude) of the matrix

    An Afine transformation multiplies a vector by a matrix and adds an offset vector, sometimes referred to as bias; like this:
    T(X) = AX + B 
    

• Eigenvectors and Eigenvalues

  So we can see that when you transform a vector using a matrix, we change its direction, length, or both. When the transformation only affects scale (in other words, the output vector has a different magnitude but the same amplitude as the input vector), the matrix multiplication for the transformation is the equivalent operation as some scalar multiplication of

Data

• type
• visualization:
• bar chart
• histogram ( frequency)
• pie chart
• scatter plots
• line charts
• Measures of Central Tendency:
  • mean,
  • 众数
• Measures of Variance:
  • percentile:
  • box & whisker chart;
  • 6 /s: 68.26% 1s, 95.45 2s, 99.73% 3s
  • z-score, z = ( x-x’) / s
• Comparing Data. discert relationship or trends

Conditional Probability and Dependence

Probability

Many of the problems we try to solve using statistics are to do with probability. For example, what’s the probable salary for a graduate who scored a given score in their final exam at school? Or, what’s the likely height of a child given the height of his or her parents?

It therefore makes sense to learn some basic principles of probability as we study statistics. Probability Basics

Let’s start with some basic definitions and principles.

An experiment or trial is an action with an uncertain outcome, such as tossing a coin.
A sample space is the set of all possible outcomes of an experiment. In a coin toss, there's a set of two possible oucomes (heads and tails).
A sample point is a single possible outcome - for example, heads)
An event is a specific outome of single instance of an experiment - for example, tossing a coin and getting tails.
Probability is a value between 0 and 1 that indicates the likelihood of a particular event, with 0 meaning that the event is impossible, and 1 meaning that the event is inevitable. In general terms, it's calculated like this:

Conditional Probability and Dependence

Events can be:

Independent (events that are not affected by other events)
Dependent (events that are conditional on other events)
Mutually Exclusive (events that can't occur together)

Binomial Variables and Distributions

A binomial variable is used to count how frequently an event occurs in a fixed number of repeated independent experiments

P(x=k)=n!/(k!*(n−k)!) *p^k *(1−p)^(n−k)

Sample and Sampling Distributions

• Confidence Intervals

• Working with Sampling Distributions

  This means we need to allow for some variation between the sample statistics and the true parameters of the full population.