STATISTICS

STATISTICS

1. ☆ S 3 胜利曙光 ☆

   • [with order, replacement] Sample distribution
     • to distinguish it with Binomial: it based on # of sample, no defined trials.
     • X1...Xn all have same distribution as the population, all independent to each other
     • Pi=[N selections] x [combinations among each selection]
       • 1. N selections: with replacement 2. combinations among unit selection: order does matter
       • steps: 1. list every combinations 2. figure out the probability of each 3. sum it
   • Combinations of n Independent Random Variables
     • (Combined V) ~ N (E(C), Var(C))
       • Normal distribution of a combined random variables.
       • E(C)=E(X1+X2+...Xn)=E(X1)+E(X2)+...E(Xn)
         • e.g. the distribution of n jams of 1kg.
         • misleading.  E(D) = E(nX) = nE(X) ≠ E(C). in words, it is even not a distribution of 'combined variables'. is the distribution of 1 jam of n kg.
       • Var(C)=Var(X1+X2+...Xn)=Var(X1)+Var(X2)+...Var(Xn)
         • Var(D) = (nX) = (n^2)(X) ≠ Var(C)
   • [和上面相反] Sample survey: deal with part of population units we desired
     • in sampling, we assume that samples are drawn from the population, sample means and population means are equal.
     • consus is really costly and inconvienent, therefore we make inferences about population with the help of samples.
     • Simple Random Sampling
       • ☆General ADV☆ 1.avoidable money wasted. 2.avoidable time wasted. 3.relatively convienent and easy to analysis as some Standard models and formulas are mature. 4.simple to apply, and carry on. 5. Truely Random:  ①Free from bias. ②each units are of equal chance being selected.
       • □General DIS-ADV□ 1.unavoidably compulsory sample frame.
       • its properties: ①each population units have equal chance of being selected. ②each be included once only. avoids repeatings.
     • Random Number Tables
       • ①ADV & DIS-ADV 和 simple random sampling 一致. ②specifically, it is not suitable and convienent to apply if sample size is large.
     • Lottery
       • ①ADV & DIS-ADV 和 SRS 一致. ②specifically, it is not convienent and suitable for large sample size (n).
     • non-random sampling
       • Quota
         • 1.uncertain interviewer: if that interviewer refuse to answer OR whatever. 2. in analogy is like to make a quick dimo.
         • ☆ADV☆  1. quickly to make a representative sample out. 2. so relatively low costs. 3. simple to apply: administration/ management is fairly easy.
         • □DIS-ADV□ 1. many errors OR uncertain are obtained in the non-random process. 2.mistakenly put interviewees in other groups. 3. introduce interviewer bias. (interviewer do sth...)
     • (no restriction on times of including the same population units) unrestricted Random sampling
     • choose which to use: ①LPS→SyS. ②LSS OR PP be ND→StS.
       • choose which to use:?? ①when population size is large, use Systematic Sampling. ②when lage sample size OR population units are naturally divided, use Stratified Sampling.
       • Systematic
         • ☆ADV☆ ①(一致). ②relatively suitable for large sample size.
         • □specific DIS-ADV□ ①not truely Random as the Ordered Sample Frame may not be set as random. and intruduces Bias.
       • Stratified
         • stratum -strata: mutually exclusive groups.
         • ☆ADV☆ specifically, ①give more representative and specific estimation. ②reflects population structure.
         • □specific DIS-ADV□ ①stratum should be defined clearly and exclusively.
     • choose which to use: ①diff SRS→NRS. ②
       • choose which to use: ①use non-random Sampling method. when it is difficult to use SRS, specifically, when (the whole population is unkown) , (the Sample Frame is hard to establish). ②
   • Significance Testing
     • CLT: Sample Means are approximately Normally Distributed
       • (X bar) ~ N( μ , (σ/√n)^2 ), μ: population mean, (σ) = (s in sample)
         • the mean and varience located in such a way is supported by Unbiased Estimator. * remember (σ/ √n), where σ is the true s.d. of population (given is Q).
       • condition: large sample size (n)
         • if sample size is large, CLT guarantees sample means are approximately normally distributed. 
         • (in Q it continues) if they (more than 1) are independent to each other. The distribution  of (combination of them) can be modeled by normal distribution.
     • Q: is it significant to reject Ho ?
   • Confidence Interval of statistic means
     • e.g. 95% CI for μ: there is 0.95 probability that the interval will contain the μ. (it does not mean that there is a probability of 0.95 that true μ lies in it. it must under some condition to have such a state.)
     • it is a range of values we fairly sure our true statistic mean lies in.
     • CI = (X bar) ± Z((1-α)/ 2) · (σ/ √n), α: given testCI
       • 1. E: margins of error. 2. (σ/√n)=SEM 3. Z: critical value.
       • width
   • Unbiased Estimators
     • as no difference in Sample Statistic and Population Parameter.
     • (补充说明) Bias is the difference in values of Estimation and Population Parameters.
     • reliability of SS to PP → estimation
       • 'point- Estimatior' applied as Sample Statistic used to estimate Population Parameters. 1. Sample statistic: *known values calculated in sample. 2. Population parameters: values calculated in population, inculdes properties.
       • (补充) 1.Estimate: is the particular value of sample statistics generated. 2. the reason we are doing this: to avoid huge calcules...
     • σ^2= (n/ n-1)·(S^2), S^2=(1/n)·∑ (Xi-barX)^2
       • 1. S^2: sample varience. S: standard deviation is called standard error in sample. 2. σ^2: population varience.
     • sample mean(barX) = population mean(μ): (∑Xi)/ n=μ
   • Regression & Correlation
     • rs = 1 - [ 6∑(di^2)/ (n)(n^2-1) ]
       • Spearman's rank correlation coefficient: measure the strength and direction of association between two ranked variables (of same size n).
     • PMCC ρ = Sxy / √(Sxx)(Syy)
       • 1. Sxy=  Sxx= Syy=
       • PMCC, Pearson product-moment correlation coefficient: measure the strength of a linear association between two variables. in appli. it attemps to draw a (straight) line of best-fit through the data of two variables.
     • (test zero correlation) Ho: rs = o OR ρ = o
   • terms
     • Population
       • a collection of set of items/people you interested in investigation.
       • Infinite
         • Countable
           • like the natural number, rational number. sth you are able to count.
         • Uncountable
           • like the real number
       • Finite
     • Sample
       • a subset of population
     • Sample Frame (X1...Xn)
       • a way to assin a number/name to each individual units. e.g. list, map, database... [how well a sample frame covers a population is important.]
     • [accurate VS t $ D massiveData] Census: deal with every population units
       • □DIS-ADV□  ①unavoidable time wasted. ②unavoidable money wasted. ③resulted in destructions. ④unavoidable massive data to deal with.
       • ☆ADV☆ it should give a completely accurate result.
     • (first and second-hand data) Primary and Secondary data
       • comparison: ☆ADV to Primary data☆ 1. known collection method and accuracy(known and manageable bias). 2. specific/target data is collected in a manageable form. 3. hard to obtain large quantity and perdictable data as internet is distrustful. □DIS-ADV to primary data□ 1. costly in time and efforts.
     • Sampling (probability) distribution: {X1, X2...Xn} of size n
       • repeat the process of taking samples, all possible samples are taken forming a random sampling distribution.
       • regulars to obey: 1. sampling distribution = probability distribution.? 2. Xi are of independent known random variables. 3. choosen [random sample distribution] has the same distribution with population.
   • Goodness of Fit (Chi-squared test)
     • calculatedχ^2=∑ [(Oi-Ei)/Ei]. testχ^2 from its FS of v & testCI
       • is a Goodness of Fit test, used with Hypothesises to find how good the static we use fits the true situation/populas.
       • Ei: np
         • specifically,in. 1. Continuous Uniform Distribution: P(a＜x＜b)=(b-a)/(Whole Range). → a proportion. Ei=(total number OR f)(P)
         • 2. Discrete
         • 3. Poisson Distribution: λ:average rate/ mean. P(X=x)=[(e^-λ)(λ^x)]/ x!.
         • 4. Binomial Distribution. P(X)=[nCr][(p)^r][(1-p)^n-r]. 不要用上面那个↑. 直接用 Cumulative Binomal Table. 减掉(你不需要的) 就好了.
         • 5. Normal Distribution:
       • v (degree of freedom)
         • v = # of expected categories - 1 - n
         • Limitation on v: we lose further n degree of freedoms due to n pieces of information to construct Ho
         • Limitation on categories: we need to combine categories to make expected frequency (Ei) ≥5
         • v = (#n - 1)(#m - 1) in Contingency table
           • is a frequency distribution table revealing two variables.

2. ☆ S 2 幸运女神 ☆

   • CONTINUOUS RANDOM VARIABLES DISTRIBUTION
     • Continuous Uniform distribution
       • X~U(a,b), P(X≤x)=F(x)=∫f(x) dx
         • a,b: boundaries
         • perhaps this is how p.d.f. of c.r.v. comes from: ∑P(x)=1=Area under a triangle. boundaries a,b. base length (b-a). height 1/(b-a) which is f(x).  and F(x) can be founded by integrate f(x).
       • properties: 1. if X~U[α,β] and Y=aX+b then Y~U[aα+b, aβ+b], 2. E(X)=(b+a)/2, Var(X)=(b-a)^2/12
         • 2. derived from general formular: a. E(X)=∫ x·f(x) dx with [a,b]. E(X)=∫ x/(b-a) dx =∫ (x^2/2)·(b-a) dx from b to a: E(X)=(b^2-a^2)/[2(b-a)] =(b+a)/2. b. Var(X)=E((x-E(X))^2) =∫ (x-E(X))^2·f(x) dx =∫ [x-(b+a)/2]^2·[1/(b-a)] dx =... =(b-a)^2/12
         • 1. derived from: consider probability function. P(y1≤y≤y2) =P(y1≤ax+b≤y2) =P((y1-b)/a≤x≤(y2-b)/a) =[(y2-b)/a-(y1-b)/a]·1/(β-α) =(y2-y1)/(β-α) and this function suggests the distribution of y.
     • Normal distribution
       • X~N(μ, σ^2), P(Z=z)=0, P(Z≤z)=∫f(z) dx= Φ(z), P(z1≤Z≤z2)=Φ(z2)-Φ(z1)
         • μ: mean σ^2: varience
       • Standarlisation: X~N(μ, σ^2) --> Y~N(0, 1^2)
         • its significance: linear transformation of a normal distribution is another normal distribution.
         • e.g. discrete to continuous to standarlised Normal: P(X>30)=P(Y>30.5)=P(Z>(30.5-μ)/σ)
   • DISCRETE RANDOM VARIABLES DISTRIBUTION
     • Binomial distribution
       • Bi means two... so this is about a distribution of a thing with 2 results.
       • conditions: trials satisfied
         • a. a set number of trials. b. trials are independent of each other. c. each trial is success or failure. d. probability of success in each trial is constant. [property of selection: order doesn't matter, with replacement]
       • Binomial Cumulative distribution table: F(x)=P(X≤x) [x matters, significance of discrete]
       • X~B(n,p), P(X=x)=(nCr)·(p^r)·((1-p)^n-r) [(p^r): what you achieved]
         • X: # of successer after n trails ~: can be modeled by B: binomial distribution n: index: # of successer p: parameter: probability of success (n-r): # of failurer
         • derived from P=N x P(s) x P(f) N: # of selections trials are independent to each other, probability can be written like that. e.g. find probability of 5heads 7tails of an unbiased dice, P=NXP(H)XP(T)=(12C5)(1/2)^5(1/2)^7
       • properties
         • E(X)=np
           • derived from: E(X)=∑ x·P(X=x) =∑ x·(nCr)(p^x)(1-p)^n-r= by calculation =np·↓ ∑ (n-1Cx-1)(p^x-1)[(1-p)^n-x] =[p+(1-p)]^n-1 =1^n-1 =1↓ =>np
         • Var(X)=np(1-p)
           • => n(p-p^2) => Var(X)∝n the more trials are the more spreading that distribution is.
     • Poission distribution [occurrences]
       • is a model for an infinite discrete, random variables satisfied some conditions.
       • X~Po(λ), P(X=x)=[(e^(-λ))(λ^x)]/x!
         • X= # of events that occur in a given interval fixed in space or time. λ (parameter) =mean=constant rate/ ()per()
         • the probability function derived by taylor series expansion: e^λ=1+λ+λ^2/2+λ^3/3!+... then mutiple e^(-λ): 1=e^-λ+(e^-λ)(λ)+(e^-λ)(λ^2)/2+... we know the condition for probability mass function of a discrete random variable, and above stuff satisfied conditions: ∑P(X=x)=1, P(X=x)≥0 [therefore we can use these values as probabilities to define a probability distribution which is named as Poission distribution of variable X with parameter λ]
       • conditions & properties : 1. conditions the variable (X) satisfied, 2. E(X)≈Var(X)≈λ [constant rate]
         • 1=> let X: # of events that occur in a given interval fixed in space or time. the event must occur: a. singly(没有两个chloe在8:05-8:10刷牙) b. independently and randomly c. at a constant rate [in the sense that # of events in a given interval is proportional to length of interval.]
       • Poission Cumulative distribution table/function: F(x)=P(X≤x), P(m≤x＜n)=P(X≤n-1)-P(X≤m-1)=F(n-1)-F(m-1)
     • Approximating Binomial distribution to Poission distribution
       • X~B(n,p), Y~Po(λ), X≈Y~Po(np=λ)
       • conditions: n is large and p is small [as a rule of thump np≤10]
         • then X can be modeled by a Poission distribution. [thump: it cannot go beyond FS]
   • APPROXIMATING Discrete random variable distribution to Continuous random variable distribution
     • Continuity Correlation
       • an adjastment made when d.r.v. be approximated to c.r.v.
       • [四舍五入] fundamentally defined (founded): X(d.r.v.). Y(c.r.v.). if X=185cm. then X= 184.5≤Y≤185.5. 对于d.r.v. ≤ 和 <是不同的含义. 对于c.r.v. ≤ 和 <含义相同.
       • derivation: if X<185cm. then X= Y< OR ≤184.5cm.
     • Normal Approximation
       • Approximating Binomial to Normal
         • X~B(n,p), Y~N(np, np(1-p)), X≈Y
         • conditions: n is large and p≈0.5
           • if they are, then the graph of binomial one is almost symmetrical. symmetrical shape is one of the big property of Normal distribution.
           • (不在考纲的知识点: the normal approximation is reasonable if both np≥10 and np(1-p)≥10)
       • Approximating Poission to Normal
         • conditions: λ is large
           • the larger λ, the more symmetrical shape is.
         • X~Po(λ), Y~N(λ, λ), X≈Y
   • Hypothesis test
     • a mathematical procedure to exam a value of population parameter proposed by null hypothesis, compared to the alternative hypothesis.
     • 1. make assumption about population parameter
       • Null hypothesis: Ho:
         • what we assume [population parameters] to be true unless otherwise.
       • Alternative hypothesis: H1
         • exactly what you think it is.[context/question dependent] [this is the conclusion you shold made if rejecting Ho] [somehow we can say Ho maximise the likelihood of get the value of static you have based on you assumption]
     • 2. collect evidence in the form of test sample
     • 3. choose a static [distribution] that is reprensentative to your assumption
     • 4. how likely it is you get the value of static you have -> from the sample based on your assumption
       • 1. find the probability of test sample which is correspondent to the assumption you made. 2. test it with 'the tool: level of significance'. A.if it is greater than significance level, then you have insufficient reason to reject Ho + conclusion made based on question. B.if it is smaller than, (you have sufficient reason to reject Ho) concluded on H1 + conclusion made on Q.
       • [test] Level of Significance
         • the probability of incorrectly rejecting Ho. OR the probability of rejecting Ho when it is in fact true.
         • [≠ 'and'] Two-tailed: half a significance level, ['or'] One-tailed
         • Critical values and Critical region
           • Critical region
             • the values of your static that collectively have a low probability of occuring. [how low: it can assess by the level of significance you chosen ] [therefore, you can say it is the range of values which would lead you to reject Ho]
           • Critical values
             • the smallest value (of a given test static) that will lead you rejecting it. [how small: assess by level of significance] [or the boundaries of critical region]
         • Actual Level of Significance
           • the actual probability we have of rejecting Ho.
         • discrete distribution: ≤α, continuous distribution: =α
     • Q: in poission when n is large it is needed to be approximated to Normal to find Cumulative Probability by applying continuity correlation

3. ☆ S 1 & S 2 最爱马拉松 ☆

   • representation & summary of data
     • Representation
       • variables collection
         • [non-numerical] qualitative
         • [numerical] quantitative
         • [particular] discrete
         • [range] continuous
       • Stem-and-leaf & Back-to-back stem-and-leaf diagrams with key
         • for discrete
         • for continuous
       • Frequency distribution, table
         • it is the different values of the variable OR the particular category arranged in order with the corresponding frequencies.
         • for discrete
         • for continuous [grouped discrete data]: intervals with frequencies
           • interval properties: 1. non-overlapping 2. cover the entire range of data 3. fewer for smaller range, vice versa
             • for continuous variables we simply dividing continuous data into classes(in your sample): properties(conditions) of interval are required since it is defined as infinite values occupy every ranges and it needed to be improved so we can see it as grouped discrete data.
         • introduction to Frequency type data
           • is the data we simply interested and studied in how frequently each of the different values of a variable OR a particular category occurs in the data set. [when process we lose the identity of individuals]
           • frequency: one of the characteristics of the groups formed by the individuals (rather than the characteristics of individuals).
       • Cumulative frequency distribution [add-up]
         • for continuous: class boundaries with cumulative frequencies
       • Grouped frequency distribution, table: class boundaries with frequency
         • -> Histogram: FD verse class boundaries
           • 1. continuous data a. frequency density:  (frequency/ class width). b. class intervals. c. class width:  (△intervals).
           • 2. groups of discrete data: a. frequency density: (frequency/ boundary width). b. class boundaries: (midpoint of every two adjacent interval values). c. boundary width: (△boundaries).
           • properties: 1. FD∝Area, 2. no space between bars, 3. ∑F=100
         • property: continuous
           • therefore we need to process discrete data into some form of continuous: we get 'class boundaries' by consider the mid-point of two adjacent interval values.
         • -> Cumulative frequency curve: cumulative frequency verse class boundaries
     • Summary
       • measures of central tendency
         • different ones are used depends on your data. 1. mean: symmetrical data. 2. median: skewed data. 3. mode: data only has several modes.
         • mean of frequency table: μ= (∑xi·fi)/ (∑fi)
         • mode: values OR intervals occurred most often
         • quartiles & percentiles: Q1=n/4, Q2=2n/4, Q3=3n/4...
           • can be found in  1.ordered list. 2.discrete frequency table. 3.grouped frequency table. 4.cumulative curve by linear interpolation.
       • measures of spread, skewness
         • (补充) 考试的目的是 characterise students, test is better if grades are quiet spread so distinguish easier.
         • range: largest - smallest
         • IQR: Q3-Q1
           • interquartile range
           • -> outliers: values outside range of (Q1-IQR) ~ (Q3+IQR)
         • s.d.: √(Var(X))
           • the reason why s.d. exist: root varience to make it dimentionally consistant (in physics). deviation is a scalar/ number, variance is a vector/ area, they cannot be equal.
         • Var(X): [∑(x-μ)^2]/n
           • 1.varience: measure of deviation. 2. why we do not simply use  [∑(x-μ)]/n as a measure of deviation: if one posi the other neg then they canceled other than showing the truth. 'square' is a tool for taking magnitude of it showing deviation.
         • shape of posi neg: μ>Q2>M
           • -> Box plots
     • Comparing data: take measures of central tendency, spread, skewness then put into context
       • interpolation: inside range, extrapolation: outside range
     • Probability
       • Classical approach: Equiprobable modal:
       • Frequensist approach: modal for discrete random variables: there is probability associated to each outcomes
         • uniform d.r.v: probability associated to each outcome is same
         • relative frequency = # of times getting Ni out of N: P(A)=Ni/N
           • e.g. P(A)=0 means the # of times you get A out of N times is 0. not mean the probability of getting A is 0.
       • probability function: P(A)
       • image representation
         • Venn: collection of outcomes
           • P(A∩B)=P(A∪B)-P(A)-P(B)
           • P(A')=1-P(A)
           • independent events: P(B/A)=P(B), P(A∩B)=P(A)·P(B)
             • it can be proved by drawing venn diagram...
           • P(B/A)=P(B∩A)/P(A)
         • Tree Diagram for independent events
           • prove of independent: P(B/R)xP(R)=[P(B∩R)/P(R)]xP(R)=P(B∩R) so P(B) is independent event.
     • Bivariate data: (x,y) may or not have a relationship
       • Coding: transform data for easier work: y=(x-a)/b
         • E(Y)=[E(X)-a]/b, Var(Y)=Var(X): 平移不计数
       • correlation: linear relationship
         • no correlation means two variables have no linear relationship, but can have non-linear relationship. e.g. if f(x) is plotted as a circle in diagram, they have no correlation, but have relationship.
         • least square line of best-fit: minimising ∑(e^2): E(Y)=a+bE(X), gradient: Sxy/Sxx
           • responsible variable(y) against explanary variable(x)
           • Q: interpretation: b: increase of [Y] to increase of [X], a: ??
         • r=Sxy/√[(Sxx)(Syy)], -1≤r≤1
           • with coding: r unchanged for new through linear transformation: movements has no effects on [n·Variation]
   • C: order matters, without replacement
   • P: order doesn't matters, without replacement
   • Random variable
     • random: probability associated to it is random. variable: it varies.
     • Discrete Random variables (d.r.v.)
       • discrete: set # of value in a given range
       • conditions: 1. the set of values it takes, 2. probability of each one by p.m.f.
       • Probability mass function (p.m.f.): f(x)=P(X=x) [piecewise function includes diff probabilities of each given x]
         • point probabilities P(X=x) determine the distribution of X.
         • properties: ∑P(X=x)=1, P(X=x)>0
         • Probability distribution table
         • -> Pin diagram []线状图]
       • properties
         • E(X): theoratical number for mean after n trials: E(X)=∑xiPi
           • derived as mean x =(∑fx)/n
           • properties: 1. E(h(x))=∑ h(xi)·Pi [E(复合函数)], 2. E(aX+b)=aE(X)+b, 3.E(X·Y)=E(X)·E(Y) if x,y are independent
         • Var(X)=E(X^2)-(E(X))^2
           • properties: Var(aX+b)=(a^2)·Var(X)+0
     • Continuous random variables (c.r.v.)
       • continuous: take any value in a given range. random: probability associated to each outcomes is random (not mean 'equal probability' to each). variables: varies
       • Probability density function (p.d.f.): f(x)=how dense the probability is.
         • properties: 1.∑f(x)=1, f(x)≥0, 2. f(x)= d/dx(F(x))
           • Q: see whether one is p.d.f.
             • steps: 1. simply drawing 2. if all f(x) are posi and total area is 1. 3. it is.
       • conditions: 1. the set of values it takes, 2. probability distribution by c.d.f. [area under p.d.f.]
         • 2=> we cannot specify probabilities (as we did for d.r.v. by using p.m.f.)
       • Cumulative distribution function (c.d.f.): F(x)=P(X≤xi)=P(X<xi), P(X=x)=0
         • xi: fixed boundary value * x: variable
         • properties: 1. F(∞)=1, 2. F(x)= ∫ f(x) dx with boundaries -∞ to xi [piecewise functions with divided intervals]
           • with[A,B], we got&nbsp; F(A)=0 F(B)=1 it is continuous and we deal with the area other than p.d.f. (don't get confused by f(A)=f(B)=0 as they referred to point probability of a continuous variable.)
           • Q: find F(x)
             • steps: 1.draw f(x) graph to simplify analysis. 2.define intervals. [it matters] 3.written down: F(x) '0,1'. f(x) '..., otherwise'. then find others by simple geometry calculations other than integration.
       • properties
         • E(X)=∫ x·f(x) dx [boundaries: b to a]
           • derived from: draw a graph you can see P=Area of small sector=δx·f(x) E(X)=∑x·[δx·f(x)]=∑x·f(x) dx since x→0, δx=dx, E(X)=∫ x·f(x) dx &nbsp;
           • properties: 1. E(X+Y)=E(X)+E(Y), 2. E(aX+b)=aE(X)+b, 3. E(h(x))=∫ h(x)·f(x) dx [h(x): given function]
             • 3=&gt; e.g. E(x^2)=∫ x^2·f(x) dx
         • Var(X)=E((x-E(X))^2) OR E(X^2)-(E(x))^2 = ∫ (x-E(X))^2·f(x) dx =∫ (x^2)·f(x) dx-(∫ x·f(x) dx)^2 with boundaries b to a
           • properties: 1. Var(aX+b)=a^2Var(X)+0, 2. Var(X+Y)=Var(X)+Var(Y) if x,y are independent
             • =&gt;1. Q: what if they are not independent to each other?
         • M: most dense: the value of x which maximise f(x): f'(x)=0
         • (Q...): the value of x at F(Q1)=0.25, F(Q2)=0.5, F(Q3)=0.75 ...
   • Normal Distribution N ( μ, σ^2 ), Z = (X-μ)/ σ
   • terms
     • probability distribution
       • is the set of possible outcomes together with their probabilities. X = (xi) with&nbsp;f(xi)