Title: Theory / Important / Math / Probability / Statistics / Cartoon+8
Tags: hasSeenConjDef examplesSwitchExample1 récupéré

Page: 75,78(84,86 at Bulmer)
Name: A binomial distribution
Chapter: 5 (6 at Bulmer)
Density function: \([0,n]\to\mathbb R\),
\(x\mapsto P^x(1-P)^{n-x}{n\choose x}\)
Mean: \(nP\)
Variance: \(nP(1-P)\)
Skewness: \(\frac{1-2P}{\sqrt{nP(1-P)} }\)
Kurtosis: \(3+\frac1{nP(1-P)}-\frac{6}n\)
Moment generating function: \((1-P+Pe^t)^n\)
Characteristic function: \((1-P+Pe^{it})^n\)
Variables: \(P\) a probability of success
\(n\in\mathbb N\)
Intuition: The number of success for \(n\) trials
Mode: \(\lfloor (n+1)P\rfloor\) (or \(\lceil (n+1)P\rceil-1\) if \(P\in(0,1)\) and the first value is an integer)
Median: \([nP]\) (ceil or floor)
Moment generating function computation: \(\sum_{k=0}^ne^{tx}p^x(1-p)^{n-x}{n \choose k}\) =
\(\sum_{k=0}^n(e^{t}p)^x(1-p)^{n-x}{n \choose k}\) =
\((e^tp+1-p)^n\)
Characteristic function computation: \(\sum_{k=0}^ne^{itx}p^x(1-p)^{n-x}{n \choose k}\) =
\(\sum_{k=0}^n(e^{it}p)^x(1-p)^{n-x}{n \choose k}\) =
\((e^{it}p+1-p)^n\)
Mean computation: \(\sum_{x=0}^nxP^x(1-P)^{n-x}{n\choose x}\) =
\(\sum_{x=0}^nxP^x(1-P)^{n-x}\frac{n!}{x!(n-x)!}\) =
\(nP\sum_{x=1}^nxP^{x-1}(1-P)^{(n-1)-(x-1)}\frac{(n-1)!}{(x-1)!((n-1)-(x-1))!}\) =
\(nP\sum_{k=1}^n{n-1\choose k-1}P^{k-1}(1-p)^{(n-1)-(k-1)}\)=
\(nP\sum_{l=0}^{n-1}{n-1\choose l}P^{l}(1-p)^{(n-1)-(l)}\)=
\(nP\left(p+(1-p)\right)^{n-1}\)=\(nP\)
Variance computation: \(X\) is the sum of \(n\) independant Bernouilli variable. Their variance is \(p(1-p)\). By sum, \(np(1-p)\)
Skewness computation: From MGF

Page: 64, 109, 111, 112
Name: Normal distribution
Chapter: 4; 7
Density function: \(\frac{e^{\frac{(x-\mu)^2}{2\sigma^2} } }{\sigma\sqrt{2\pi} }=\frac{e^{\frac12\left(\frac{x-\mu}{\sigma}\right)^2 } }{\sigma\sqrt{2\pi} }\)
Mean: \(\mu\)
Variance: \(\sigma^2\)
Skewness: 0
Kurtosis: 3
Moment generating function: \(e^{t\left(\mu+\frac{\sigma^2t}{2}\right)}\)
Characteristic function: \(e^{t\left(i\mu -\frac{\sigma^2t}{2}\right)}\)
Variables: \(\mu\in\mathbb R\)
\(\sigma\in\mathbb R^{+}\)
Cumulative function: \(\frac12\left[1+erf\left(\frac{x-\mu}{\sigma\sqrt2}\right)\right]\)
Standard deviation: \(\sigma\)
Nth moment: \(\sigma^n\Pi_{i=0}^{r/2}(2i)+1\)
Kind: Cartoon: ch 5. P 81
Bulmer Ch 4, 7. p 64, 109, 111, 112

Page: 64, 109, 111, 112
Name: Standard Normate distribution
Chapter: 4; 7
Density function: \(\frac{e^{\frac{x^2}{2} } }{\sqrt{2\pi} }\)
Mean: 0
Variance: 1
Kurtosis: 3
Moment generating function: \(e^{\left(\frac{t^2}{2}\right)}\)
Characteristic function: \(e^{t\left(-\frac{t}{2}\right)}\)
Intuition: Normal distribution with mean 0 and standard variation 1
Cumulative function: \(\frac12\left[1+erf\left(\frac{x}{\sqrt2}\right)\right]\)
Nth moment: \(\Pi_{i=0}^{n/2}(2i)+1\)

Name: The variable which follows the \(t\) distribution with \(f\) degrees of freedom
Definition: \(\mathcal Z\) standard normal variate
Page: 132
Chapter: 8
Section: \(t\) Distribution
Definition2: \(\mathcal Y\) follows a \(\xi^2\) distribution with \(f\) degrees of freedom
Conjdef: \(\frac{\mathcal Z}{\sqrt{\mathcal Y/f}}\)
Definition3: variables are independant
Denoted by: \(\mathcal T\)

Definition: \(E((x-\mu)^2)\)
Page: 56
Chapter: 4
Section: measure of dispersion
Variables: \(f\) a probability density function
Notation: \(S^2\)

Name: Data analysis
Definition: Gathering display and summary of data
Page: 4
Chapter: 1

Name: Probability
Definition: The law of chance
Page: 4
Chapter: 1

Name: Statistical inference
Definition: Science of drawing statistical conclusions from specific data using a knowledge of probability.
Page: 4
Chapter: 1

Question: Statistician's raw material
Answer: data
Page: 8
Chapter: 2

Name: Dot plot
Definition: \(n\) dots over \(i\), when the value \(i\) appears \(n\) times in the sequence.
Page: 8
Chapter: 2
Variables: \(X\) a subset of reals
\((x_i)\in X\) a sequence of data

Name: The frequency of \((x_i)\) in \(X\)
Definition: The number of elements of \(x_i\) belonging to \(I\)
Page: 10
Chapter: 2
Variables: \(X\) a subset of reals
Let \((x_i)\in X\) a sequence of data
\(I\) an interval

Name: The relative frequency of \((x_i)\) in \(X\)
Definition: The frequency, divided by the length of the sequence
Page: 10
Chapter: 2
Variables: \(X\) a subset of reals
Let \((x_i)\in X\) a sequence of data
\(I\) an interval

Question: Assume you want to partition the range of the \(x_i\)'s into successive interval, to show data about the frequency of their values. What rules should you follow ?
Answer: Use interval of equal length
Page: 10
Chapter: 2
Variables: \(X\) a subset of reals
Let \((x_i)\in X\) a sequence of data
Answer2: Midpoints of intervals are conveniant numbers
Answer3: the number of interval is roughly proportional to the number of data

Name: An histogram
Definition: Representing a rectangle above \(I_i\) whose height is proportional to the frequency of \((x_i)\) in \(I_i\)
Page: 11
Chapter: 2
Variables: \(X\) a subset of reals
\((x_i)\in X\) a sequence of data
\((I_i)\) a partition of \(X\) into intervals

Name: Steam and leaf diagram
Definition: A column of the values \(x_i/10\).
For each \(x_i/10\), writes \(x_i\mod 10\) on its right, in increasing order.
Page: 12
Chapter: 2
Variables: \(X\) a subset of reals
Let \((x_i)\in X\) a sequence of data

Name: The mean
Definition: \(\frac{\sum_{i=1}^nx_i}{n}\)
Page: 15
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data
Notation: \(\overline x\)
Name2: The average

Name: The median
Definition: A value \(m\) such that there is as many \(x_i\) above \(m\) than \(x_i\) below \(m\)
Page: 17
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in\mathbb R\) a sequence of data
Intuition: Middle data point

Name: The first quartile
Definition: A value \(m\) such that there is three time as many \(x_i\) above \(m\) than \(x_i\) below \(m\)
Page: 19
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data
Notation: \(Q_1\)

Name: The third quartile
Definition: A value \(m\) such that there is three time as many \(x_i\) below \(m\) than \(x_i\) above \(m\)
Page: 20
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data
Notation: \(Q_3\)

Name: Interquartile range
Definition: \(Q_3-Q_1\)
Page: 20
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data
Abbreviation: IQR

Name: Box and whiskers plot
Definition: Draw the real line.
Page: 20
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data
Definition2: Draw lines above the median and each quartile. As rectangle
Conjdef: Obtained by
Definition3: Draw outlier
Definition4: Draw whiskers out to the farthest points that are not outliers

Name: An outlier
Definition: A point which is more that \(1.5\) IQR below (above) from an the first (third) quartil
Page: 20
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data

Name: Sample variance
Definition: \(\frac{\sum_{i=1}^n(x_i-\overline x)^2}{n-1}\)
Page: 22
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data
Notation: \(S^2\)
Intuition: Average (squared) distance from the mean

Name: Standard deviation
Definition: square root of the variance
Page: 23
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data
Abbreviation: \(S\)

Name: The \(z\)-score of \(z_i\)
Definition: \(\frac{x_i-\overline x}{S}\)
Page: 23
Chapter: 2
Variables: \((x_i)_{1\le i\le n}\in \mathbb R\) a sequence of data
\(i\in[1,n]\)
Name2: the standardized score of \(z_i\)
Intuition: The number of standard deviation above or below the mean

Name: A random experiment
Definition: Process of observing the outcome of a chance event
Page: 30
Chapter: 2
Example: For coin tosses: recording its outcome

Name: Elementary outcome of \(R\)
Definition: All possible results of \(R\)
Page: 30
Chapter: 2
Variables: \(R\) a random experiment
Example: For coin tossing: Head and tail

Name: Sample space of \(R\)
Definition: The set or collection of all elementary outcomes of \(R\)
Page: 30
Chapter: 2
Variables: \(R\) a random experiment

Name: The probability distribution on \((O_i)_i\)
Definition: The likelihood of \(O_i\) to occurs
Page: 32
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
Abbreviation: \(P(O_i)\)
Typ: Non-negative
Property: sums to 1

Name: Classical (historic) approach to probability
Definition: all elementary outcomes have the same probability
Page: 35
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes

Name: Relative frequency
Definition: The proportion of time an event occurs in the long run
Page: 35
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes

Name: Personal probability
Definition: Assessment of an outcome's likelihood
Page: 35
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes

Name: Objectivist
Definition: Someone using classical or frequency definition of probability
Page: 35
Chapter: 2

Name: A subjectivist
Definition: Someone using personal definition of probability
Page: 35
Chapter: 2
Name2: A Bayesian

Name: An event
Definition: A set of elementary outcome
Page: 36
Chapter: 2
Example: Rolling two dices: having a 7

Name: The probability of \(E\)
Definition: The sum of the probability of the elements of \(E\)
Page: 36
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E\) an event

Name: \(E\) and \(F\)
Definition: The elementary outcomes belonging to \(E\) and to \(F\)
Page: 36
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events

Name: \(E\) or \(F\)
Definition: The set of elementary outcomes belonging to \(E\) or to \(F\)
Page: 36
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events

Name: Not \(E\)
Definition: The set of elementary outcomes not belonging to \(E\)
Page: 36
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events

Smaller0: \(P(E\text {or} F)+P(E\text {and} F)\)
Greater0: \(P(E)+P(F)\)
Page: 36
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events
Smaller to greater: =
Name: Addition rule

Smaller0: \(E\) and \(F\) are mutually exclusive
Greater0: equals \(P(E)+P(F)\)
Page: 39
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events
Prefix Greater: \(P(E\text {or} F)\)
Name: Addition rule

Question: \(P(\text{not }E)\)
Answer: equals \(1-P(E)\)
Page: 39
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events
Name: Substraction rule
Extra: Probability

Name: Probability of \(A\) given \(C\)
Definition: \(\frac{P(A\text{ and }C)}{P(C)}\)
Page: 41
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events
Notation: \(P(A\mid C)\)

Question: \(P(E\mid E)\)
Answer: 1
Page: 41
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events

Smaller0: \(P(E\text{ and }F)\)
Greater0: \(P(E\mid F)P(F)\)
Page: 42
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(E,F\) events
Smaller to greater: =
Name: Multiplication rule
Smallest: \(P(F\mid E)P(E)\)

Name: Independent events
Definition: \(P(E,F)=P(E)P(F)\)
Page: 43
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
Definition2: \(P(E)=P(E\mid F)\)
Conjdef: Two events such that, equivalently
Extra: using H and I
Intuition: Events such that the the truth of one has no effect of the truth of the other

Question: \(P(O_j\mid I_i)\)
Answer: equals \(\frac{P(I_i\mid O_j)P(O_j)}{\sum_{j'} P(I_i\mid O_{j'})P(O_{j'})}\)
Page: 49
Chapter: 2
Variables: \(R\) a random experiment
\(O_1,O_2\) two elementary outcome
\(I_1,I_2\) two other elementy outcomes
Assume that \(P(I_i\mid O_j)\), \(P(I_i\mid \text{not }O_j)\) and \(P(O_i)\) are known
Construction: \(\frac{P(O_j,I_i)}{P(I_i)}=\frac{P(O_jI_i)}{\sum_j P(I_i,O_j)}=\frac{P(I_i\mid O_j)P(O_j)}{\sum_{j'} P(I_i\mid O_{j'})P(O_{j'})}\)

Name: Random variable
Definition: A function whose domain is \(\{O_i\mid i\}\)
Page: 54
Chapter: 4
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
Example: Given a student, its height, weight, grades, age...
Example2: For dices, the sum of the results
Denoted by: \(\mathbf{X}\)

Name: The probability of \(x\)
Definition: \(\sum_{O_i\mid P(O_i)=x}P(O_i)\)
Page: 55, 60
Chapter: 4
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(X:\{O_i\}\to V\) a random variable
\(x\in V\)
Notation: \(P(x)\)
Notation2: \(P(X=x)\)
Notation3: \(n_x\)

Name: The probability distribution of \(X\)
Definition: [$]\defFun{V}{[0,1]}{x}{P(X=x)}[/$]
Page: 55
Chapter: 4
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(X:\{O_i\}\to V\) a random variable

Name: The histogram of \(X\)
Definition: An horizontal line labelled by the elements of \(V\). Over each \(v\in V\), a rectrange of height proportional to \(P(v)\).
Page: 55
Chapter: 4
Variables: \(R\) a random experiment
\(O_1,\dots,O_n\) the elementary outcomes
\(X:\{O_i\}\to V\) a random variable

Name: Sample properties
Definition: Properties related to data
Page: 59
Chapter: 4
Context: statistics

Name: Model properties
Definition: Properties related to probability distribution
Page: 59
Chapter: 4
Context: Statistics
Name2: Population properties

Name: The mean of \(p\)
Definition: \(\sum_{x\in V}xp(x)\)
Page: 60
Chapter: 4
Variables: \(p:V\to\mathbb R\) a probability distribution
Notation: \(E[X]\)
Name2: The expected value of \(X\)
Denoted by: \(\mu\)
Assuming: \(V\subseteq\mathbb R\)

Name: The variance of \(p\)
Definition: \(\sum_{x\in V}(x-\mu)^2p(x)\) with \(\mu\) the mean
Page: 62
Chapter: 4
Variables: \(p:V\to\mathbb R\) a probability distribution
Notation: \(\sigma^2\)
Assuming: \(V\subseteq\mathbb R\)

Name: The standard deviation of \(p\)
Definition: square root of the variance of \(p\)
Page: 62
Chapter: 4
Variables: \(p:V\to\mathbb R\) a probability distribution
Notation: \(\sigma\)
Assuming: \(V\subseteq\mathbb R\)

Name: Pseudo-random numbers
Definition: A sequence of numbers, seeming random, but generated by an algorithm
Page: 65
Chapter: 4

Name: A probability density
Definition: A function \(\mathbb R\to[0,1]\) whose integral is 1
Page: 65
Chapter: 4

Name: The probability that \(X\) is in \([a,b]\)
Definition: [$]\int_a^bf(x)\dd x[/$]
Page: 65
Chapter: 4
Variables: \(f\) probability density
\(a\le b\)
Notation: \(Pr(a\le x\le b)\)
Intuition: The area under the curve between \(x\)-values \(a\) and \(b\)

Name: The mean of \(f\)
Definition: [$]\int_{-\infty}^\infty x f(x)\dd x[/$]
Page: 65
Chapter: 4
Variables: \(f\) probability density
\(a\le b\)

Name: The variance of \(f\)
Definition: [$]\int_{-\infty}^\infty (x-\mu)^2 f(x)\dd x[/$]
Page: 65
Chapter: 4
Variables: \(f\) probability density
\(a\le b\)

Question: \(E(\lambda+\sum\lambda_i X_i)\)
Answer: equals \(\lambda+\sum\lambda_i E(X_i)\)
Page: 69
Chapter: 4
Variables: \(X_1,\dots, X_r\) random variables
\(\lambda_i\) reals

Question: \(\sigma^2(\lambda+\sum\lambda_i X_i)\)
Answer: equals \(\sum\lambda_i^2\sigma^2(X_i)\)
Page: 71
Chapter: 4
Variables: \(X_1,\dots, X_r\) random variables
\(\lambda_i\) reals
Assuming: each \(X_i\) independant

Name: A binomial random variable
Definition: A random variable taking exactly two values. Called Success and failure
Page: 74
Chapter: 5

Name: Bernouilli trial
Definition: where each success have same probability
Page: 74
Chapter: 5
Definition2: each trials are independant
Conjdef: A sequence of binomial random variables

Question: \(Pr(X=k)\)
Answer: equals \({n\choose k}p^k(1-p)^{n-k}\)
Page: 76
Chapter: 5
Variables: \(X\) a Bernouilli trial with probability \(p\) of success, and \(n\) trials
\(k\le n\)

Name: \(n\) choose \(k\)
Definition: \(\frac{n!}{k!(n-k)!}\)
Page: 76
Chapter: 5
Variables: \(k\le n\)
Notation: \({n\choose k}\)
Intuition: Number of way of choosing \(k\) elements in a set of \(n\) elements

Name: Pascal's triangle
Definition: the entry of line \(i\) and column \(j\) is \(i\choose k\)
Page: 77
Chapter: 5
Variables: \(n\in\mathbb N\)
Definition2: the entry \((0,0)\) is 1.
The entry on line \(i\) and column \(j\) \((i,j)\) is the sum of the entries on \((i-1,j)\) and \((i-1,j-1)\), assumed to be \(0\) if it is not defined
Conjdef: A table, such that, equivalently

Question: The binomial distribution of \(n\) trials with probability \(p\) is approximated by
Answer: the normal curve with \(\mu=np\) and \(\sigma^2=np(1-p)\)
Page: 82
Chapter: 5
Extra: \(\sigma\) in the book is probably a typo

Name: \(n\) factorial
Definition: \(1*\dots*n\)
Page: 631
Chapter: XI
Notation: \(n!\)
Extra: Cartoon statistics, ch5 P 76
Cracking the code interview ch XI, P 631
Intuition: Number of way to list \(n\) distinct things

Page: 83
Chapter: 5
Notation: Fuzzy central Limit Theorem
Intuition: data influenced by many small and unrelated things can be approximated by a normally distributed distribution

Name: The z-transformation
Definition: The operation \(x\mapsto\frac{x-\mu}{\sigma}\) with \(\mu\) the mean and \(\sigma\) the standard deviation
Page: 83
Chapter: 5
Context: Statistics
Intuition: Approxating \(x\) by a standard normal variable

Smaller0: \(Pr(a\le X\le b)\)
Greater0: \(F(\frac{b-\mu}{\sigma})-F(\frac{a-\mu}{\sigma})\), with \(F\) the density curve of the standard normate variable
Page: 83
Chapter: 5
Variables: \(x\) variables normally distributed
\(a,b\in\mathbb R\)
Smaller to greater: equals

Name: Continuity correction
Definition: Approximating \(Pr(a\le X\le b)\) by \(Pr\left(\frac{a-\frac 12-np}{\sqrt{np(1-p)}}\le Z\le\frac{a+\frac 12-np}{\sqrt{np(1-p)}}\right)\), when \(np\ge 5\) and \(n(1-p)\ge5\)
Page: 87
Chapter: 5
Variables: \(p\in[0,1]\)
\(X\) the number of successes in \(n\) independant Bernouilli trials with probability \(p\)
Intuition: an adjustment that is made when a discrete distribution is approximated by a continuous distribution