PROBABILITY FOR ENGR APPL
PROBABILITY FOR ENGR APPL ECSE 4500
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This 70 page Class Notes was uploaded by Miss Damien Crooks on Monday October 19, 2015. The Class Notes belongs to ECSE 4500 at Rensselaer Polytechnic Institute taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/224787/ecse-4500-rensselaer-polytechnic-institute in ELECTRICAL AND COMPUTER ENGINEERING at Rensselaer Polytechnic Institute.
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Date Created: 10/19/15
C jx f y PD Qinju aw H F 1 F F 5 c us ltH be in Statistical concepts and methods provide insights into the behavior of many phenomena that we encounter in life and thus are used in man39 fields of s ecialization in humanities sciences and engineering This discipline is concerned with how to make intelligent judgments and informed decisions in the presence of uncertainty and variation Without uncertainty or variation there would be no need for statistical methods Provides methods for organizing and summarizing olata drawing conclusions based on information contained in the data in addition it provides us with suggestiOns for how to design efficient ways for collecting data ucpcllulllg Ull vvnat inferenCes we are interested in now we move to agree on some terminology The is everything you wish to study eg all RPI students who have come to campus before noon today A Is used to represent a cnaracterlstlc of each member of the population eg female vs mal g orstanding freshmansenior of the individuals in the RPI population mentioned above A is a stud of the entire o ulation eg1 we may have a population on which We study election results because we do not have timeresources to conduct a census as on election day lfwe have all data for all eligible voters then we havea census eg239for the preceding example all RPI students A Simple Random Sample is a sample that has been selected in such a way that all members of the population have an e ua chance of bein included in the sample eg how to do this for the preceding RPl students example is the difference between the behavior of the entire population and a sample of that population 1 z The amount of refers to how different the members of the population are from each other with regard to the variable being studied variance measures the variation number of members of the population It is referred to as 1 is referred to F s is a sample which does not re resent the o ulation Q5 is data that can take on only certain values These values are often integers or whole numbers 39 IS aata that can taKe on any one of an infinite number of possible values over an interval on He nunmer line how many real numbers are there between 01 These values are most often the result of measurement allow you A F r uu39 a y w 39 UU i 710 U m h is a deduction of a F 7 16 quot3 FQ39Q w Mafw mg u S n I m s m c n o c r o S e c n m o e f I m w a r d o t s U w m la 5 conclusion to summarize data An We will use Probability theory to calculate the likelihood of observing or selecting a particular sample from a I o ulation E M F A F 3 H v 1 392 F w Q n u 7 I or tt etgLUJ ltM99 is a table containing each category value or class of values that a variable might take and the number of times that each one occurs in the data eg try an example by hand for 20 students with integer grades 1 10 53 lea715 7 HZ 60 Ar 39p p The iwgeiatwe of a classification is the number of times an observation falls into that classification represented as a mum It can be expressed as a fraction decimal or percentage Faxquot 92 I n Winn W3 a In quot quotJquot39 x9751 We 1F The cowlm I m c 0 L a class is the sum of the relative frequencies of all classes at or below that class represented as a portion of the total number of observations It can be expressed as a fraction decimal or percentage A Bar Chart represents the frequency or relative frequency from the table in the form of a rectangle or bar Class Year of Students in Introductory Statistics 12 quot 10 quot 8 Number of 6 Students Freshmen Sophomore Junior Senior Class A is a picture a bar chart of a frequency distribution in which the yaxis represents the frequencies and the xaxis represents the observed measurements The xaxis is divided into sections consistent with the class 0 bin width and constitute the base for a rectangle The height of the rectangle along the y axis is consistent with the frequency of the observations in the class or bin Each bin can be labeled with its center point or its bounds H 01 H 0 Frequency gt O E a 3 U39 3 q C 2 4 E d D 70 90 110 130 150 170 190 210 230 250 Compressive strength psi Example of a Histogram 200 250 Strength Example of a Cumulative Distribution Plot I I 50 100 1 f U1 M m EC ECU 6 Elm Number u glasses and Class Interval for Continuous Data classesn Manua bin calculation based on problem objectives ExcelMinitab will apply a default bin ranges unless you Specify one V100 10 classes 8291510 314 cell width A Pie Chart represents data in the form of slices or sections of a circle Each slice re resents a cateor and the size of the slice is proportional to the relative frequency of the category Class Year of Introductory Statistics tudents Freshman 14 senior 36 Jynior homore 7o J It Q 397 7 jfx r 3 39 X r quotA quotH u I r um 5 39D Hir K1310 f r Q UDD LL nger W tt 1 puma LLQECDHkE g l P 9 In a each observation is plotted as a point on a single horizontal axis The axis is scaled so that each of the data points can be located uniquely on the axis When there is more than one observation with the same value the points are stacked on top of each other i is 39 L N Each number in the data set should consist of at least two digits 21g 1 x UCV L 3 Aiu Divide each number in to two arts stem and leaf Heftm m 3 L firemanaim 6 UL 1 Frequency 1 l 1 2 3 3 6 8 553 Lp gto1 The of a set of data describes how the data are spread out around the center with respect to the symmetry or skewness of the data The of a set of data describes how the data are spread out around the center with respect to the smoothness and magnitude of the variation When data are not evenly spreau out on either side of the center then we refer to the distribution as being Histogram of Bimodal Data 339 5 61391131395139715 23912393 X A Is a numerical descriptor that is and is used to describe the sample Statistics are usually represented by Roman letters eg Xbar and s A is a numerical descriptor that is used to Parameters are usually represented by Greek letters eg p and o The is the center of balance of a set of data and is found by adding up all of the data values and dividing by the number of observations The is represented by the Greek letter u The is the value of the middle observation in an ordered set of data Consider Sampr uata taken from a certain population Let Xi be the ith obs for i12 n where n is the number of observations in the sample The sample mean is Histogram of Bimodal Data 339 5 61391131395139715 23912393 X Q A is the difference between the maximum and minimum observations in the sample V F DA A was 7 F A I The sf Is the average of the squared deviations of the daa values from the sample mean 39 T h e 1 557 rm m N 13 3767 P1457 mm 7 a v 7 6177 W W 19 L94 D J L A L 2 Q3 17 941 U U KQJ L94 J KQ 1 UJ U 3 the positive square root of the sample variance With a pOpulation ofsize N the mean and variance are computed using The says that for a bell shaped symmetric distribution about 68 of all data values are within one standard deviation of the mean about 95 of all observations are within two standard deviations of the mean almost all more than 99 of the observations are within three standard deviations of the mean A zscore measures the number or standard deviations that a data value is from the mean The Pth Percentile of a data set is the value that has p of the data at or below it so thei99th percentileseparates the highest 1 from the remaining lower 99 The of a value is the percentage of observations that are at or below the value of interest Pth I ercentile a at or below I 7 Percentile rank be2N b number 0 data values below the value of interest e the number of other observations equal to the value first set e0 to understand the equation N population size p Trimmed Mean truncates p of the distribution from each end of it Provides a measure that is not as extre me as the mean or median Demonstration of Percentile Calculations ECSE4500 Exam scores 20 values 82 65 91 83 75 80 74 63 72 79 93 55 64 84 80 90 81 73 50 95 Scores in sequence 5055636465727374757980808182838490919395 The 90th and 10th percentiles 91 55 Percentile rank of 90 1 60220 30 Percentile rank of 72 50220 25 The 10 trimmed mean 63646572737475798080818283849091l16 7788 vs mean value of 7645 Since it s an even number of observations the median is obtained as 79802 795 a The is the value in the sample that has 25 of the data at or below it The is the value in the sample that has 75 of the data at or below it A is a graphical display that uses summary statistics to display the distribution of a set of data A Interquartile Range IQR is the difference between the third and first quartiles Q3 Q1 The Inner Fences of a boxplot are located at Q1 15 IQR and 3903 15 IQR The Outer Fences of a boxplm are located at Q1 3 IQR and 3903 3 IQR Illustrate a box plot for problem 156 using Minitab Box Plot Q1 3QR Q33IQR 39 Q115QR Q1 Mediaquot Q3 Q15IQR Iquot IQR gtI ECSE 4500 Probability for Engineering Applications Fall 2008 Fields of Events A collection of sets Fsatisfying three axioms 1 Q5 and 2 e F 2 Aand BeFABandAUBeF 3 A67A 6f Probability space Q F P In words probability P measures all events sets in the field F consisting of the subsets of the sample space 2 whose elements are the outcomes of a random experiment An example Pick at random a numbered ball from 12 balls in an urn The balls are labeled 1 through 12 U2w with A gquot m Pgquot 2 given and for any event A we nave PA Z Pm all geA Note A can be any subset of the 212 subsets of Q Let A126 and B349 then AUB1 2 3 97 AB3456 ABC 12 and ABC 127812 To be specific let PKG 2112 to get PA1 2 PB712 PAUB912 etc Conditional probability Consider two events A and B in the event field Fof sample space 2 Take PBgtO Then PM BPABPB Note If we fix the event B and then let A range over all the events in F then P B is a valid probability measure ie it satisfies the three axioms of probability Q Can you show this When PAgtO also we can write PBA also PBAPBAPA These two conditional probabilities are related PA PB APBAPAB PM BPB thus PB APA BPBlPlAl which is quite a helpful consequence ofthe basic definition of conditional probability Binary symmetric channel BSC Consider a random experiment with sample space 2 consisting of the ordered pairs XY with Xthe input to a communications channel and Ythe corresponding output We take a binary alphabet so thatXand Ytake on only the values 0 and 1 Hence the sample space consists of only four outcomes Q00O11011 Consider first the event corresponding to input XO D n in hi v n XO we can write it as X00001 and similarly for the event X1 X1lt1ogtlt11gt 1 To model a balanced source we take PXOPX112 Next we introduce the conditional probabilities PYXO and PYX1 to model the channel part of our experiment When event XO occurs there only two outcomes 00 and 01 possible the first of which 00 corresponds to a correct output while the latter 01 outcome indicates an error We set PO1XOp and P00XO1p with 0ltplt1 For the symmetric case we would have P10X1p and P01X11p also 0 139 0 We can pictorially represent these conditional probabilities 1 1 with the BSC diagram 1D Independence Def Two events A and B are independent if PABPAPB If PAgtO then this is equivalent to PBIAPB Note that independence is very different from mutual exclusivity or disjointness in fact if ABq the A and B cannot be independent unless PA or PB equal zero Total Probability Theorem Let A1 A2 Anbe a mutually exclusive and collectively exhaustive decomposition of Q ie Al mAj unless i j and UAI Q i1 with PAl gt 0 for each i then for any event B we have PM PB I Aim1i Proof of Total Probability can In mmmH304j HO zi Z PB AiPAl as was to be shown 391 Eg for binary comm channel We want to comf ute P Y 07 Letl1 X 0 andl2 X 1 and n 2 Then we have a valid decomposition So PY 0 PY 0A1PA1PY 0 I Ann12 09gtlt0501gtlt 05 05 Similarly PY 1 PY 0C1 PY 0 210 05 05 Bayes Theorem We combine Total Probability with PB APA BPBPA to get For any event A J in a valid decomposition lUL B be an event With PB gt u then we have PB I AJPA 72 EM 41PM PA B Cont d BSC example We want to compute PX O Y 1 Let11 lX V wqu lX2 wunzg Then we have a valid decomposition not B lY 1 Plugging mu we formula we get H Al I B Z PBA1PA1 PB I A PM PB AZIPIAZI PY 1 X 0PX 0 PY 1 01x05 05 201PX0Y1 ECSE 4500 Probability for Engineering Applications Fall 2008 Lmiwg 1 0 What is probability A theory used to model the chance occurrence of events 0 What is statistics The estimation ofthe parameters in probabalistic models eg estimating means and variances Areas of application Games of chance Turbulence in fluid flow Noise in communications Reliability in computer systems Networking paCKet loss scneoullng etc Statistical signal processing Image and video compression Internet page statistics 4 types of probability A Probability as intuition it will probably rain Classical theory ratio of favorable outcomes of experiment to total number possible outcomes Relative frequency of occurrence Von Mises Axiomatic theory Kolmogorov the one we study Classical example Throw two fair six sided dice and record the outcome as an ordered pair dvdz 11 12 13 14 15 16 Note ThIs set of 21 22 23 all outcomes 31 32 is called the 41 I 51 56 Samp 6 space 61 65 66 Note that all these 36 outcomes are equally likely Classical theory cont w 0 What is the probability of getting 34 Ans nF n total total number of outcomes rim p with number of favorable outcomes nF 1 and 36 So p136 Reason All the outcomes are equally likely here Q What happens when they are not Classical theory cont Consider that we now enquire about the sum of the top faces of the two dice Later we will call this a random variable We have S d1 d2 millnu 3 4 5 6 7 8 valuesofsmapped 4 5 6 7 8 over the outcomes gt 9 Notezvaluesofsare 5 6 7 8 9 10 notequallylikely 6 7 8 9 10 11 7 s 9 1o 11 12 n 5 1156 F n 36 total Relative frequency approach Repeat the experiment n times Let nE number of times event E occurs S t e pE lim 71 00 n Problem How do we know the limit will exist Experimental evidence suggests it will at least sometimes No theory backing though Axiomatic theory Based on set theory events are sets 0 Probability is just a measure of these sets Must satisfy three fundamental axioms iPLEJ 2 u for all events E iiPQ 1 where Q is the certain event iiiPA U B PA PB 11 A m B 2 where is the null event
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