Formulas FROM LECTURE 01 TO 22 W X. d n. fx f. Arslan Latif (mt ) & Mohsin Ali (mc ) Mean: Weighted Mean: Mean Deviation: Ungroup Data


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1 1 Formulas FROM LECTURE 01 TO Mea: fx f Weighted Mea: X w W X i i Wi Mea Deviatio: Ugroup Data d M. D Group Data fi di M. D f d ( X X ) Coefficiet of Mea Deviatio: M. D Coefficiet of M. D(for mea) Mea M. D Coefficiet of M. D(for media) Media Stadard Deviatio: Ugroup Data ( X X ) S. D Group Data f ( X X ) S. D f
2 Shortcut Formula for Ugroup data x x S. D Shortcut Formula for Group Data fx fx S. D f f CoEfficiet of Stadard Deviatio: S. D X Variace: Ugroup Data Variace ( X X ) Group Data Variace ( ) f X X f CoEfficiet of Variatio:. C. V S D 100 X Media: + 1 Media item h Media l + c f Harmoic Mea: Ugroup Data X 1 x Group Data f X f x th
3 3 Average Speed: Total Distace Travelled Average Speed Total Time Take Distace Time Speed Distace Speed Time Distace Speed Mode: Time ( fm f 1) X l + h ( fm f 1) + ( fm f ) Empirical Relatio betwee Mea, Media ad Mode: Mode 3Media Mea Rage: m 0 Rage X X MidRage: Mid Rage X 0 + X m Mid Quatile Rage: Mid Rage Q1 + Q3 Coefficiet of Dispersio: Coefficiet of Dispersio X X m m X + X 0 0 Geometric Mea: Ugroup Data log X G. M ati log Group Data f log X G. M ati log f
4 4 Quartile Deviatio: Q3 Q1 Q. D Q Q Co efficiet of Q. D Q + Q th + 1 Q1 item 4 h Q1 l + c f 4 h 3 Q3 l + c f 4 Skewess: SK Mea Mode S. D Pearso s Coefficiet of Skewess 3( Mea Media) SK S. D Bowley s coefficiet of Skewess Q1 + Q3 ( Media) SK Q Q Kurtosis: K Q. D P P Momets: Ugroup Data ( X i X ) m1 0 ( X i X ) m 3 ( X i X ) m3 Group Data m 3 f ( X X ) Momet Ratios: i f ( m3 ) m4 ad b ( m ) ( m ) b 1 3 3
5 5 Equatio of a Straight Lie Y a + bx Stadard Error of Estimate: s yx Y a Y b XY Normal Equatios: Y a + b X XY a X + b X Correlatio: r ( X X )( Y Y ) ( X X ) ( Y Y ) PEARSON S COEFFICIENT OF CORRELATION r XY ( X )( Y ) / [ X ( X ) / ][ Y ( Y ) / ] Liear Correlatio of Coefficiet: r xy ( x)( y) x ( x) y ( y) Regressio Lie of Y o X: xy ( x)( y) byx x ( x) Divisio of Circle: Cell Frequecy 360 Total Frequecy Biomial Coefficiet:( ( C r ) ( a + b) a b r 0 r r r
6 6 1. Cumulative Law: A B B A A B B A. Associative Law: ( A B) C A ( B C) ( A B) C A ( B C) 3. Distributive Law: A (B C) (A B) (A C) A (B C) (A B) (A C) 4. Idempotet Law: A A A A A A 5. Idetity Law: A S S A S A A φ A A φ φ0 6. Complemetatio Law: A A S A A φ ( A) A S φ φ S 7. De Morga s Law: ( ) ( ) A B A B A B A B Rule of Permutatio:! r! ( ) Rule of Combiatio:! r! r! ( )
7 7 Probability Formula: m P(A) No. of Favourable Outcomes P(A) Total No. of Possible Outcomes Mutually Exclusive Evets: If A ad B mutually exclusive evets, P(A B) P(A) + P(B) Complemetatio Law of Probability: P(A) 1 P( A) Additio Law: P(A B) P( A) + P( B) P( A B) Coditioal Probability: P(A/B) P(B/A) P(A B) P(B) P(A B) P(A) Multiplicatio Law: P(A B) P(A)P(B/A) Baye s Theorem: P(A B) 1 P( A ). P( B A ) 1 1 P( A ). P( B A ) + P( A ). P( B A ) 1 1 Computatio of Stadard Deviatio Probability Distributio: X P( X ) XP( X ) S. D.( X ) P( X ) P( X ) S. D.( X ) X P( X ) XP( X ) As we kow, P( X ) 1 Coefficiet of Variatio: 100 µ
8 8 Total Cards 5 Diamod 13 Heart 13 Club 13 Spades 13 Face Cards 1 Quee 4 Jack 4 Kig 4 Jocker 4 FROM LECTURE 3 TO 45 Mathematical Expectatios: E( X ) x f ( x ) i 1 i i E(X) is also called the mea of X ad is usually deoted by the letter µ. If the values are equally likely: 1 E( X ) X i If X is a discrete radom variable ad if a ad b are costats, the: E ( ax + b) ae( X ) + b If µ is a Populatio Mea the: E ( X µ ) E( X ) [ E( X )] Shortcut Formula for the Variace i Mathematical Expectatios: E( X ) [ E( X )] Skewess of Probability: µ 3 β1 3 µ Kurtosis of Probability: 4 µ β1 µ Mea o Probability: E( X ) XP( X )
9 9 If the Parameters i distributio are a ad b the µ a + b Mea of the Populatio Distributio E( X ) xf ( x) µ Mea of the SAMPLIN DISTRIBUTION of X µ E( X ) x f ( x) x µ µ x Variace of Probability: Var( X ) X P( X ) [ XP( X )] Expected value of Variace Var( X ) or E( X µ ) ( x µ ) f ( x) If the Parameters i distributio are a ad b the ( b a) 1 Variace of the Populatio Distributio x f ( x) x f ( x) Variace of the SAMPLIN DISTRIBUTION of X Ugroup Data X X ( X ) Group Data X X f ( X ) X f ( X ) I case of samplig with replacemet: x I case of samplig without replacemet from a fiite populatio: x N N 1 i Fiite Populatio Correctio (fpc) N N 1
10 10 Biomial Distributio: x P( X x) p q x x IN CASE OF NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION f ( x) x x x p q Joit Probability Fuctio: f ( x, y ) kth Momet: m ' r X r i Origi of Radom Variable µ k E( Xk) X k f ( x) Mea of Radom Variable µ k i E( X µ ) k ( X µ ) k f ( x) Chebychev s Iequality: i I the case of a Discrete Probability Distributio 1 P( µ k < X < µ + k ) 1 k E µ µ [( X ) ] E( X ) MARGINAL PROBABILITY: Margial Probability fuctio of X g( x ) f ( x, y ) i i j j 1
11 11 Margial Probability fuctio of Y m h( y ) f ( x, y ) P( Y y ) i i j j i 1 Coditioal Probability Fuctio: Coditioal Probability fuctio for X f ( x y ) P( X x Y y ) i j i j Coditioal Probability fuctio for Y f ( y x ) P( Y y X x ) j i j i HYPER GEOMETRIC PROBABILITY DISTRIBUTION: P( X x) k N k ( x )( x ) N ( ) Where N umber of uits i the populatio, umber of uits i the sample, ad k umber of successes i the populatio. HYPER GEOMETRIC DISTRIBUTION: Mea of Hyper Geometric Distributios k µ N Variace of Hyper Geometric Distributio k N k N N N N 1 Biomial Probability Distributio: Lim p 0 b(x;,p) x 0,1,,..., e.1788 µ p e x! µ x µ UNIFORM DISTRIBUTION: A radom variable X is said to be uiformly distributed if its desity fuctio is defied as
12 1 1 f ( x), a x b b a Poisso Process Formula: e P( X x) ( t) x! λ t x λ P( X x) e µ x µ x! Where λ average umber of occurreces of the outcome of iterest per uit of time, t umber of timeuits uder cosideratio, ad x umber of occurreces of the outcome of iterest i t uits of time. NORMAL DISTRIBUTION: 1 f ( x) e π Where : π or 7 e x µ Momet Ratios i Probability: The momet ratios of the ormal distributio come out to be 0 ad 3 respectively: µ 0 β µ ( ) 4 4 µ 3 β 3 µ ( ) Stadardizatio Formula: Z X µ I case of X X µ x Z x
13 13 Proportio of Successes i the Populatio: X p or p N X p Proportio of Success X Number of successes i the populatio Samplig distributio of p SAMPLING DISTRIBUTION OF DIFFERENCES BETWEEN MEANS: µ µ µ x x 1 ad µ x1 x 1 1 x x 1 ad x x 1 ( x1 x) f ( x1 x) d f ( d) df ( d) x1 + x Z ( X1 X ) ( µ 1 µ ) SAMPLING DISTRIBUTION OF p µ p p f ( p ) Stadard Deviatio of the Samplig Distributio: p q p1 + p p q Modified Formula for the Sample Variace: S ( x x) 1
14 14 Relative Efficiecy: E f Var( T ) Var( T ) 1 Differece betwee the Meas of Two Populatios: s s ( x1 x ) ± zα / CONFIDENCE INTERVAL FOR A POPULATION PROPORTION (P): p ± z α / p (1 p ) Where p Proportio of "success" i hte sample Sample size z α / 1.96 for 95% cofidece.58 for 99% cofidece CONFIDENCE INTERVAL FOR P1P: p1(1 p1) p(1 p ) ( p1 p) ± zα / + 1 SAMPLE SIZE FOR ESTIMATING POPULATION PROPORTION: p z α / pq ɵ tdistribution: 1 x f ( x) 1 1 v + v vβ, Variace of the tdistributio v v for v > ( v+ 1)/
15 15 FDistributio: [ ] Γ ( v + v ) / ( v / v ) x < x < 1 Γ( v / ) Γ ( v / ) 1 + v x / v v1 / ( v1 /) ( ), 0 ( v + v )/ f x [ ] 1 1 Mea of the Fdistributio: For v > v v Variace of the Fdistributio: For v > 4 v ( v + v ) ( ) ( 4) 1 v1 v v Mea Square: Mea Square Sum of Squares Degrees of Freedom Least Sigificat Differece: LSD t α /,( v) ( MSE) r
16 16 Defiitios FROM LECTURE 01 TO Statistics: Statistics is that sciece which eables to draw coclusios about various pheomea oe the basis of real data collected o sample basis. OR Statistics is a sciece of facts ad figures. INFERENTIAL STATISTICS: That brach of Statistics which eables us to draw coclusios or ifereces about various pheomea o the basis of real data collected o sample basis. Data: A well defied collectio of objects is kow as data. Qualitative data: Data that are labels or ames used to idetify a attribute of each elemet. Qualitative data may be oumeric or umeric. Qualitative variable: A variable with qualitative data. Quatitative data: Data that idicate how much or how may of somethig. Quatitative data are always umeric. Variable: A measurable quatity which ca vary from oe idividual or object to aother is called a variable. Nomial Scale: The classificatio or groupig of observatios ito mutually exclusive qualitative categories is said to costitute a omial scale e.g. studets are classified as male ad female. Ordial Scale: It icludes the characteristic of a omial scale ad i additio has the property of orderig or rakig of measuremets e.g. the performace of studets ca be rated as excellet, good or poor. Iterval Scale: A measuremet scale possessig a costat iterval size but ot true zero poit is called a Iterval Scale. Ratio Scale: It is a special kid of a iterval scale i which the scale of measuremet has a true zero poit as its origi.
17 17 Biased Errors: A error is said to be biased whe the observed value is cosistetly ad costatly higher or lower tha the true value. Ubiased Errors or Radom Errors: A error, o the other had, is said to be ubiased whe the deviatios, i.e. the excesses ad defects, from the true value ted to occur equally ofte. Primary Data: The data published or used by a orgaizatio which origially collected them are called primary data thus the primary data are the first had iformatio collected, complied, ad published by a orgaizatio for a certai purpose. Secodary Data: The data published or used by a orgaizatio other tha the oe which origially collected them are kow as secodary data. DIRECT PERSONAL INVESTIGATION: I this method, a ivestigator collects the iformatio persoally from the idividuals cocered. Sice he iterviews the iformats himself, the iformatio collected is geerally cosidered quite accurate ad complete. INDIRECT INVESTIGATION: Sometimes the direct sources do ot exist or the iformats hesitate to respod for some reaso or other. I such a case, third parties or witesses havig iformatio are iterviewed. Populatio: The collectio of all idividuals, items or data uder cosideratio i statistical study is called Populatio. Sample: A sample is a group of uits selected from a larger group (the populatio). By studyig the sample it is hoped to draw valid coclusios about the larger group. OR Sample is that part of the Populatio from which iformatio is collected. SAMPLING FRAME: A samplig frame is a complete list of all the elemets i the populatio. Samplig Error: The samplig error is the differece betwee the sample statistic ad the populatio parameter. NoSamplig Error: Such errors which are ot attributable to samplig but arise i the process of data collectio eve if a complete cout is carried out. Samplig with replacemet: Oce a elemet has bee icluded i the sample, it is retured to the populatio. A previously selected elemet ca be selected agai ad therefore may appear i the sample more tha oce. Samplig without replacemet: Oce a elemet has bee icluded i the sample, it is removed from the populatio ad caot be selected a secod time.
18 18 Stadard error: The stadard deviatio of a poit estimator. AND The degree of scatter of the observed values about the regressio lie measured by what is called stadard deviatio of regressio or stadard error of estimate. Samplig Uit: The uits selected for samplig. A samplig uit may iclude several elemets. NONRANDOM SAMPLING: Noradom samplig implies that kid of samplig i which the populatio uits are draw ito the sample by usig oe s persoal judgmet. This type of samplig is also kow as purposive samplig. Quota Samplig: Quota samplig is a method of samplig widely used i opiio pollig ad market research. Iterviewers are each give a quota of subjects of specified type to attempt to recruit for example, a iterviewer might be told to go out ad select 0 adult me ad 0 adult wome, 10 teeage girls ad 10 teeage boys so that they could iterview them about their televisio viewig. RANDOM SAMPLING: The theory of statistical samplig rests o the assumptio that the selectio of the sample uits has bee carried out i a radom maer. By radom samplig we mea samplig that has bee doe by adoptig the lottery method. Simple radom samplig: Fiite populatio: a sample selected such that each possible sample of size has the same probability of beig selected. Ifiite populatio: a sample selected such that each elemet comes from the same populatio ad the elemets are selected idepedetly. Pie Chart: Pie Chart cosists of a circle which is divided ito two or more mars i accordace with the umber of distict classes that we have i our data. SIMPLE BAR CHART: A simple bar chart cosists of horizotal or vertical bars of equal width ad legths proportioal to values they represet. MULTIPLE BAR CHARTS: This kid of a chart cosists of a set of grouped bars, the legths of which are proportioate to the values of our variables, ad each of which is shaded or colored differetly i order to aid idetificatio. CLASS BOUNDARIES: The true class limits of a class are kow as its class boudaries. HISTOGRAM: A histogram cosists of a set of adjacet rectagles whose bases are marked off by class boudaries alog the Xaxis, ad whose heights are proportioal to the frequecies associated with the respective classes. FREQUENCY POLYGON: A frequecy polygo is obtaied by plottig the class frequecies agaist the midpoits of the classes, ad coectig the poits so obtaied by straight lie segmets.
19 19 FREQUENCY CURVE: Whe the frequecy polygo is smoothed, we obtai what may be called the frequecy curve. CUMULATIVE FREQUENCY DISTRIBUTION: As i the case of the frequecy distributio of a discrete variable, if we start addig the frequecies of our frequecy table columwise, we obtai the colum of cumulative frequecies. AVERAGES (I.E. MEASURES OF CENTRAL TENDENCY): A sigle value which iteded to represet a distributio or a set of data as a whole is called a average. It is more or less a cetral value aroud which the observatios ted to cluster so it is called measure of cetral tedecy. Sice measure of cetral tedecy idicate the locatio of the distributio o X axis so it is also called measure of locatio. The Arithmetic, Geometric ad Harmoic meas Are averages that are mathematical i character, ad give a idicatio of the magitude of the observed values. The Media Idicates the middle positio while the mode provides iformatio about the most frequet value i the distributio or the set of data. THE MODE: The Mode is defied as that value which occurs most frequetly i a set of data i.e. it idicates the most commo result. DOT PLOT: The horizotal axis of a dot plot cotais a scale for the quatitative variable that we wat to represet. The umerical value of each measuremet i the data set is located o the horizotal scale by a dot. GROUPING ERROR: Groupig error refers to the error that is itroduced by the assumptio that all the values fallig i a class are equal to the midpoit of the class iterval. Ogive: A graph of a cumulative distributio. Dispersio: The variability that exists betwee data set. Rage: The rage is defied as the differece betwee the maximum ad miimum values of a data set. The coefficiet of variatio: The coefficiet of variatio expresses the stadard deviatio as the percetage of the arithmetic mea. Quartiles: Quartiles are those three quatities that divide the distributio ito four equal parts. Quartile Deviatio: The quartile deviatio is defied as half of the differece betwee the first ad third quartiles.
20 0 Quatiles: Collectively the quartiles, the deciles, percetiles ad other values obtaied by equall subdivisio of the data are called quatiles. Percetiles: Percetiles are those iety ie quatities that divide the distributio ito hudred equall parts Absolute measure of dispersio: A absolute measure of dispersio is oe that measures the dispersio i terms of the same uits or i the square of uits, as the uits of the data. Relative measure of dispersio: Relative measure of dispersio is oe that is expressed i the form of a ratio, coefficiet of percetage ad is idepedet of the uits of measuremet. COEFFICIENT OF QUARTILE DEVIATION: The Coefficiet of Quartile Deviatio is a pure umber ad is used for COMPARING the variatio i two or more sets of data. Mea Deviatio: The mea deviatio is defied as the arithmetic mea of the deviatios measured either from the mea or from the media, all deviatios beig couted as positive. Variace: Variace is defied as the square of the stadard deviatio. Stadard Deviatio: Stadard Deviatio is defied as the positive square root of the mea of the squared deviatios of the values from their mea. Fiveumber summary: A exploratory data aalysis techique that uses the followig five umbers to summarize the data set: smallest value, first quartile, media, third quartile, ad largest value. Momets: Momets are the arithmetic meas of the powers to which the deviatios are raised. Correlatio: Correlatio is a measure of the stregth or the degree of relatioship betwee two radom variables. OR Iterdepedece of two variables is called correlatio. OR Correlatio is a techique which measures the stregth of associatio betwee two variables. RANDOM EXPERIMENT: A experimet which produces differet results eve though it is repeated a large umber of times uder essetially similar coditios is called a Radom Experimet. SAMPLE SPACE: A set cosistig of all possible outcomes that ca result from a radom experimet (real or coceptual), ca be defied as the sample space for the experimet ad is deoted by the letter S. Each possible outcome is a member of the sample space, ad is called a sample poit i that space.
21 1 SIMPLE & COMPOUND EVENTS: A evet that cotais exactly oe sample poit is defied as a simple evet. A compoud evet cotais more tha oe sample poit, ad is produced by the uio of simple evets. MUTUALLY EXCLUSIVE EVENTS: Two evets A ad B of a sigle experimet are said to be mutually exclusive or disjoit if ad oly if they caot both occur at the same time i.e. they have o poits i commo. EXHAUSTIVE EVENTS: Evets are said to be collectively exhaustive, whe the uio of mutually exclusive evets is equal to the etire sample space S. EQUALLY LIKELY EVENTS: Two evets A ad B are said to be equally likely, whe oe evet is as likely to occur as the other. I other words, each evet should occur i equal umber i repeated trials. Skewess: Skewess is the lack of symmetry i a distributio aroud some cetral value (mea, media or mode).it is thus the degree of a symmetry. Kurtosis: Kurtosis is the degree of peakess of a distributio usually take relative to a ormal distributio. MOMENTS: A momet desigates the power to which deviatios are raised before averagig them. Probability: Probability is defied as the ratio of favorable cases over equally likely cases. SUBJECTIVE OR PERSONALISTIC PROBABILITY: Probability i this sese is purely subjective, ad is based o whatever evidece is available to the idividual. It has a disadvatage that two or more persos faced with the same evidece may arrive at differet probabilities. Coditioal Probability: The probability of the occurrece of a evet A whe it is kow that some other evet B has already occurred is called the coditioal probability. MULTIPLICATION LAW: The probability that two evets A ad B will both occur is equal to the probability that oe of the evets will occur multiplied by the coditioal probability that the other evet will occur give that the first evet has already occurred. INDEPENDENT EVENTS: Two evets A ad B i the same sample space S, are defied to be idepedet (or statistically idepedet) if the probability that oe evet occurs, is ot affected by whether the other evet has or has ot occurred, Baye s theorem: A method used to compute posterior probabilities. RANDOM VARIABLE: Such a umerical quatity whose value is determied by the outcome of a radom experimet is called a radom variable.
22 Discrete radom variable: A radom variable that may assume either a fiite umber of values or a ifiite sequece of values. Set: A set is ay welldefied collectio or list of distict objects, e.g. a group of studets, the books i a library, the itegers betwee 1 ad 100, all huma beigs o the earth, etc. SUBSET: A set that cosists of some elemets of aother set, is called a subset of that set. For example, if B is a subset of A, the every member of set B is also a member of set A. If B is a subset of A, we write: (B A) PROPER SUBSET: If a set B cotais some but ot all of the elemets of aother set A, while A cotais each elemet of B, the the set B is defied to be a proper subset of A. Uiversal Set: The origial set of which all the sets we talk about, are subsets, is called the uiversal set (or the space) ad is geerally deoted by S or Ω. VENN DIAGRAM: A diagram that is uderstood to represet sets by circular regios, parts of circular regios or their complemets with respect to a rectagle represetig the space S is called a Ve diagram, amed after the Eglish logicia Joh Ve ( ). The Ve diagrams are used to represet sets ad subsets i a pictorial way ad to verify the relatioship amog sets ad subsets. UNION OF SETS: The uio or sum of two sets A ad B, deoted by (A B), ad read as A uio B, meas the set of all elemets that belog to at least oe of the sets A ad B, that is (A B) INTERSECTION OF SETS: The itersectio of two sets A ad B, deoted by A B, ad read as A itersectio B, meas that the set of all elemets that belog to both A ad B. OPERATIONS ON SETS: The basic operatios are uio, itersectio, differece ad complemetatio. SET DIFFERENCE: The differece of two sets A ad B, deoted by A B or by A (A B), is the set of all elemets of A which do ot belog to B. DISJOINT SETS: Two sets A ad B are defied to be disjoit or mutually exclusive or ooverlappig whe they have o elemets i commo, i.e. whe their itersectio is a empty set i.e. A B φ Cojoit Sets: Two sets A ad B are said to be cojoit whe they have at least oe elemet i commo. COMPLEMENTATION: The particular differece S A, that is, the set of all those elemets of S which do ot belog to A, is called the complemet of A ad is deoted by A or by Ac.
23 3 POWER SET: The class of ALL subsets of a set A is called the Power Set of A ad is deoted by P(A). For example, if A {H, T}, the P(A) {φ, {H}, {T}, {H, T}}. CARTESIAN PRODUCT OF SETS: The Cartesia product of sets A ad B, deoted by A B, (read as A cross B ), is a set that cotais all ordered pairs (x, y), where x belogs to A ad y belogs to B. This set is also called the Cartesia set of A ad B set of A ad B, amed after the Frech mathematicia Ree Descartes ( ). TREE DIAGRAM: The tree is costructed from the left to the right. A tree diagram is a useful device for eumeratig all the possible outcomes of two or more sequetial evets. PERMUTATION: A permutatio is ay ordered subset from a set of distict objects. OR A arragemet of all or some of a set of objects i a defiite order is called permutatio. COMBINATION: A combiatio is ay subset of r objects, selected without regard to their order, from a set of distict objects. OCCURRENCE OF AN EVENT: A evet A is said to occur if ad oly if the outcome of the experimet correspods to some elemet of A. PARTITION OF SETS: A partitio of a set S is a subdivisio of the set ito oempty subsets that are disjoit ad exhaustive, i.e. their uio is the set S itself. CLASS OF SETS: A set of sets is called a class. For example, i a set of lies, each lie is a set of poits. DISTRIBUTION FUNCTION: The distributio fuctio of a radom variable X, deoted by F(x), is defied by F(x) P(X < x). FROM LECTURE 3 TO 45 Chebychev s Iequality: If X is a radom variable havig mea µ ad variace > 0, ad k is ay positive costat, the the probability that a value of X falls withi k stadard deviatios of the mea is at least. CONTINUOUS RANDOM VARIABLE: A radom variable X is defied to be cotiuous if it ca assume every possible value i a iterval [a, b], a < b, where a ad b may be ad + respectively. JOINT DISTRIBUTIONS: The distributio of two or more radom variables which are observed simultaeously whe a experimet is performed is called their JOINT distributio.
24 4 BIVARIATE PROBABILITY FUNCTION: The joit or bivariate probability distributio cosistig of all pairs of values (xi, yj). MARGINAL PROBABILITY FUNCTIONS: The poit to be uderstood here is that, from the joit probability fuctio for (X, Y), we ca obtai the INDIVIDUAL probability fuctio of X ad Y. Such idividual probability fuctios are called MARGINAL probability fuctios. CONDITIONAL PROBABILITY FUNCTION: Let X ad Y be two discrete r.v. s with joit probability fuctio f(x, y). The the coditioal probability fuctio for X give Y y, deoted as f(x y). INDEPENDENCE: Two discrete r.v. s X ad Y are said to be statistically idepedet, if ad oly if, for all possible pairs of values (xi, yj) the joit probability fuctio f(x, y) ca be expressed as the product of the two margial probability fuctios. COVARIANCE OF TWO RANDOM VARIABLES: The covariace of two r.v. s X ad Y is a umerical measure of the extet to which their values ted to icrease or decrease together. It is deoted by XY or Cov (X, Y), ad is defied as the expected value of the product. BINOMIAL DISTRIBUTION: The biomial distributio is a very importat discrete probability distributio. It was discovered by James Beroulli about the year PROPERTIES OF A BINOMIAL EXPERIMENT: Every trial results i a success or a failure. The successive trials are idepedet. The probability of success, p, remais costat from trial to trial. The umber of trials,, is fixed i advaced. Biomial probability distributio: A probability distributio showig the probability of x successes i trials of a biomial experimet. Biomial probability fuctio: The fuctio used to compute probabilities i a biomial experimet. Biomial experimet: A probability experimet havig the followig four properties: cosists of idetical trials, two outcomes (success ad failure) are possible o each trial, probability of success does ot chage from trial to trail, ad the trials are idepedet. PROPERTIES OF A BINOMIAL EXPERIMENT: Every item selected will either be defective (i.e. success) or ot defective (i.e. failure) Every item draw is idepedet of every other item The probability of obtaiig a defective item i.e. 7% is the same (costat) for all items. (This probability figure is accordig to relative frequecy defiitio of probability. Hyper Geometric Probability fuctio: The fuctio used to compute the probability of x successes i trials whe the trials are depedet.
25 5 PROPERTIES OF HYPERGEOMETRIC EXPERIMENT: The outcomes of each trial may be classified ito oe of two categories, success ad failure. The probability of success chages o each trial. The successive trials are ot idepedet. The experimet is repeated a fixed umber of times. Hyper Geometric Distributio: There are may experimets i which the coditio of idepedece is violated ad the probability of success does ot remai costat for all trials. Such experimets are called hyper geometric experimets. PROPERTIES OF THE HYPERGEOMETRIC DISTRIBUTION: If N becomes idefiitely large, the hyper geometric probability distributio teds to the BINOMIAL probability distributio. The above property will be best uderstood with referece to the followig importat poits: There are two ways of drawig a sample from a populatio, samplig with replacemet, ad samplig without replacemet. Also, a sample ca be draw from either a fiite populatio or a ifiite populatio. Poisso Probability Distributio: A probability distributio showig the probability of x occurreces of a evet over a specified iterval of time or space. Poisso Probability Fuctio: The fuctio used to compute Poisso probabilities. PROPERTIES OF POISSON DISTRIBUTION: It is a limitig approximatio to the biomial distributio, whe p, the probability of success is very small but, the umber of trials is so large that the product p µ is of a moderate size; a distributio i its ow right by cosiderig a POISSON PROCESS where evets occur radomly over a specified iterval of time or space or legth. POISSON PROCESS: It may be defied as a physical process govered at least i part by some radom mechaism. NORMAL DISTRIBUTION: A cotiuous radom variable is said to be ormally distributed with mea μ ad stadard deviatio if its probability desity fuctio is give by (Formula of Normal Distributio) THE STANDARD NORMAL DISTRIBUTION: A ormal distributio whose mea is zero ad whose stadard deviatio is 1 is kow as the stadard ormal distributio. THE PROCESS OF STANDARDIZATION: I other words, the stadardizatio formula coverts our ormal distributio to the oe whose mea is 0 ad whose stadard deviatio is equal to 1. SAMPLING DISTRIBUTION: The probability distributio of ay statistic (such as the mea, the stadard deviatio, the proportio of successes i a sample, etc.) is kow as its samplig distributio.
26 6 CENTRAL LIMIT THEOREM: The theorem states that: If a variable X from a populatio has mea μ ad fiite variace, the the samplig distributio of the sample mea X approaches a ormal distributio with mea μ ad variace / as the sample size approaches ifiity. Samplig Distributio: A probability distributio cosistig of all possible values of a sample statistic. SAMPLING DISTRIBUTION OF THE SAMPLE PROPORTION: I this regard, the first poit to be oted is that, wheever the elemets of a populatio ca be classified ito two categories, techically called success ad failure, we may be iterested i the proportio of successes i the populatio. POINT ESTIMATION: Poit estimatio of a populatio parameter provides as a estimate a sigle value calculated from the sample that is likely to be close i magitude to the ukow parameter. UNBIASEDNESS: A estimator is defied to be ubiased if the statistic used as a estimator has its expected value equal to the true value of the populatio parameter beig estimated. CONSISTENCY: A estimator θˆ is said to be a cosistet estimator of the parameter θ if, for ay arbitrarily small positive quatity e. EFFICIENCY: A ubiased estimator is defied to be efficiet if the variace of its samplig distributio is smaller tha that of the samplig distributio of ay other ubiased estimator of the same parameter. METHODS OF POINTESTIMATION: The Method of Momets The Method of Least Squares The Method of Maximum Likelihood These methods give estimates which may differ as the methods are based o differet theories of estimatio. THE METHOD OF LEAST SQUARES: The method of Least Squares, which is due to Gauss ( ) ad Markov ( ), is based o the theory of liear estimatio. It is regarded as oe of the importat methods of poit estimatio. METHOD OF MAXIMUM LIKELIHOOD: This method was itroduced i 19 by Sir Roald A. Fisher ( ).The mathematical techique of fidig Maximum Likelihood Estimators is a bit advaced, ad ivolves the cocept of the Likelihood Fuctio. HYPOTHESISTESTING: It is a procedure which eables us to decide o the basis of iformatio obtaied from sample data whether to accept or reject a statemet or a assumptio about the value of a populatio parameter. NULL HYPOTHESIS: A ull hypothesis, geerally deoted by the symbol H0, is ay hypothesis which is to be tested for possible rejectio or ullificatio uder the assumptio that it is true.
27 7 ALTERNATIVE HYPOTHESIS: A alterative hypothesis is ay other hypothesis which we are willig to accept whe the ull hypothesis H0 is rejected. It is customarily deoted by H1 or HA. TYPEI AND TYPEII ERRORS: O the basis of sample iformatio, we may reject a ull hypothesis H0, whe it is, i fact, true or we may accept a ull hypothesis H0, whe it is actually false. The probability of makig a Type I error is covetioally deoted by α ad that of committig a Type II error is idicated by β. TESTSTATISTIC: A statistic (i.e. a fuctio of the sample data ot cotaiig ay parameters), which provides a basis for testig a ull hypothesis, is called a teststatistic. PROPERTIES OF STUDENT S tdistribution: i) The tdistributio is bellshaped ad symmetric about the value t 0, ragig from to. ii) The umber of degrees of freedom determies the shape of the tdistributio. PROPERTIES OF FDISTRIBUTION: 1. The Fdistributio is a cotiuous distributio ragig from zero to plus ifiity.. The curve of the Fdistributio is positively skewed. ANALYSIS OF VARIANCE (ANOVA): It is a procedure which eables us to test the hypothesis of equality of several populatio meas. EXPERIMENTAL DESIGN: By a experimetal desig, we mea a pla used to collect the data relevat to the problem uder study i such a way as to provide a basis for valid ad objective iferece about the stated problem. The pla usually icludes: The selectio of treatmets, whose effects are to be studied, The specificatio of the experimetal layout, ad The assigmet of treatmets to the experimetal uits. SYSTEMATIC AND RANDOMIZED DESIGNS: I this course, we will be discussig oly the radomized desigs, ad, i this regard, it should be oted that for the radomized desigs, the aalysis of the collected data is carried out through the techique kow as Aalysis of Variace. THE COMPLETELY RANDOMIZED DESIGN (CR DESIGN): A completely radomized (CR) desig, which is the simplest type of the basic desigs, may be defied as a desig i which the treatmets are assiged to experimetal uits completely at radom, i.e. the radomizatio is doe without ay restrictios. THE RANDOMIZED COMPLETE BLOCK DESIGN (RCB DESIGN): A radomized complete block (RCB) desig is the oe i which The experimetal material (which is ot homogeeous overall) is divided ito groups or blocks i such a maer that the experimetal uits withi a particular block are relatively homogeeous. Each block cotais a complete set of treatmets, i.e., it costitutes a replicatio of treatmets. The treatmets are allocated at radom to the experimetal uits withi each block, which meas the radomizatio is restricted. (A ew radomizatio is made for every block.)the object of this type of arragemet is to brig the variability of the experimetal material uder cotrol.
28 8 Least Squares Method: The method used to develop the estimated regressio equatio. It miimizes the sum of squared residuals (the deviatios betwee the observed values of the depedet variable, yi, ad the estimated values of the depedet variable, yi) Level of Sigificace: Level of sigificace of a test is the probability used as a stadard for rejectig ull hypothesis Ho whe Ho is assumed to be true. The level of sigificace acts as a basis for determiig the critical regio of the test. THE LEAST SIGNIFICANT DIFFERENCE (LSD) TEST: Accordig to this procedure, we compute the smallest differece that would be judged sigificat, ad compare the absolute values of all differeces of meas with it. This smallest differece is called the least sigificat differece or LSD. PROPERTIES OF THE CHISQUARE DISTRIBUTION: The ChiSquare (χ) distributio has the followig properties: 1. It is a cotiuous distributio ragig from 0 to +.The umber of degrees of freedom determies the shape of the chisquare distributio. (Thus, there is a differet chisquare distributio for each umber of degrees of freedom. As such, it is a whole family of distributios.). The curve of a chisquare distributio is positively skewed. The skewess decreases as ν icreases. ASSUMPTIONS OF THE CHISQUARE TEST OF GOODNESS OF FIT: While applyig the chisquare test of goodess of fit, certai requiremets must be satisfied, three of which are as follows: 1. The total umber of observatios (i.e. the sample size) should be at least 50.. The expected umber ei i ay of the categories should ot be less tha 5. (So, whe the expected frequecy ei i ay category is less tha 5, we may combie this category with oe or more of the other categories to get ei 5.) 3. The observatios i the sample or the frequecies of the categories should be idepedet. DEGREES OF FREEDOM: As you will recall, whe discussig the tdistributio, the chisquare distributio, ad the Fdistributio, it was coveyed to you that the parameters that exists i the equatios of those distributios are kow as degrees of freedom. P value: The pvalue is a property of the data, ad it idicates how improbable the obtaied result really is. LATEST STATISTICAL DEFINITION: Statistics is a sciece of decisio makig for goverig the state affairs. It collects aalyzes, maages, moitors, iterprets, evaluates ad validates iformatio. Statistics is Iformatio Sciece ad Iformatio Sciece is Statistics. It is a applicable sciece as its tools are applied to all scieces icludig humaities ad social scieces.
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