Introductory Statistics for Engineers
Introductory Statistics for Engineers STAT 224
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Date Created: 09/17/15
Stat 224 Handout 6 TA Shubing Wang Oct 26 2005 1 Poisson Distribution A random variable X is said to have a Poisson distribution with parameter gt 0 if the pmf of X is iAm m 5 7 x012 13 So where is e A from Since we can have Maclaurin expansion of it A A2 ZAm e 1i2 x0 i mi So we can have for Poisson pmf 00 2pm 1 m0 The mean and variance of Poisson rV X is Suppose 71 a 7 0 then 51 mp H 2990 What a Poisson pmf looks like Powsson pmmm a 5 i 016 e 0147 7 012 r A 01 r lt 23 Q008 7 006 e 0047 7 002 i i t i 0 5 10 15 x Matlab code for the plot X 015 y poisspdfx5 P10txy Exercise 380 10 computers experience CPU failure Consider a sample of 10000 computer a The expected value and SD of the number of computers in the sample having CPU failure So the experiment is binomial with n 10000 and p 001 so EX np 10 and SDX sqrtVX npq 3161 b The approximate probability that more than 10 sampled computers have CPU failure X has ap proximately a Poisson distribution with 10000001 Exercise 388 Poisson process with rate 04 05 An arriving vehicle have no violation with probability 05 a The probability that 10 arrived with no violation P10 arrive and No violation PNo violation 10 arriveP 10 arrive 0510 p10 10 By Matlab gtgt poisspdf 10 10 ans O 12 5 l b y 2 10 the probability that y arrive and 10 have no violation Py arrive and 10 No violation P10 No violation y arriveP y arrive 1310 y 05 py 10 bionomail and poisson c The probability that ten noviolation cars arrive during the hour The summation of probability inbfory1011 Exercise 3103 40 carry single spikelets 60 carry double spikelets A seed with single spikelts produce an ear with 29 While with double produce with 26 of time a The probability that exactly ve of these seeds carry a single spikelet and produce an ear Witha single spikelets Let event C seed carries single spikelets and event P seed produces ears with single spikelets Then PP O C PPlC x PC 2940 116 Let X the number of seeds out of the 10 selected that meet the condition P O C Then X N Bin10 116 b The probability that exactly 5 of the ears produced by these seeds have single spikelets PP PP O C PP O C Y N Bin107 PP where y the number out of the 10 seeds that meet condition P 2 Continuous RV De nition a rV X is said to be continuous if 3A lt B such that RangeX A7 B The pdf of X x is a positive function such that b Pa m g b 1 Examples of continuous rV s Uniform rV and exponontial rV Example 44 Exercise 45 X the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is 0 otherwise 02fzdz1 b The probability that the lecture ends within 1 min of the end of the hour kmz 0 m 2 a Find the value of k Note that 1 PO x l A m im c The lecture continues beyond the hour for between 60 and 90 sec Pl x g 15 d The prob of at least 90 sec PX 2 15 STAT 224 TA Lane Burgette Sepi 2728 DISCUSSION 3 1 Probability ll Axioms o For any event A PA 2 0 0 138 l o If A1 A2 An are mutually disjoint then PA1 UAQ u UAn ZPAZ i391 2i Conditional Probability o For any two events A and B with PB gt 0 the conditional probability of A given that B has occurred is de ned PAB LAEB P o The Multiplication Rule PA B PAlB PB o The Law of Total Probability Let A1 Ak be mutually exclusive and exhaustive eventsi Then for any other event B 133 PBlA1PA1 PBlAkPAk k ZPBlAiPAi o Bayes7 Theorem Let A1 Ak be collection of k mutually exclusive and exhaustive events with PAi gt 0 for i l kl Then for any other event B for which PB gt 0 391k PAj B L NB 7 M B 221 PltBAigt PltAigt 3 Independence 0 Two events A and B are independent if PA B PA and are dependent otherwise 0 A and B are independent if and only if PA B PA PB 0 Events A1 An are mutually independent if for every k k 23 n and every subset of indices i17i27 39 quotJim PAi1 Ai2 Aik PAi139PAi2quotquot39PAik 2 Exercises 227 An academic department with ve faculty members Anderson Box Cox Cramer and Fisher must select two of its members to serve on a personnel review committee Because the work will be timeconsuming no one is anxious to serve so it is decided that the representative will be selected by putting ve slips of paper in a box mixing them and selecting two a What is the probability that both Anderson and Box will be selected b What is the probability that at least one of the two members whose name begins with C is selected STAT 224 TA Lane Burgette Sep 2728 C If the ve faculty members have taught for 3 6 7 10 and 14 years respectively at the university what is the probability that the two chosen representatives have at least 15 years7 teaching experience at the university Let us consider rolling two sixsided fair dice Let A be the event that the sum of the two rolls is 8 Let B be the event that the sum of the two rolls is 7 Let C be the event that the rst roll is a 3 a Are the events A and C dependent or independent b Are the events B and C independent Let us say that we interview a group of people in a cancer hospital Suppose that a people are smokers with lung cancer 12 people are nonsmokers with lung cancer 5 are smokers without lung cancer and d of the people are nonsmokers with no lung cancer Express all the relevant conditional probabilities in terms of a b c and d 234 A production facility employs 20 workers on the day shift 15 workers on the swing shift and 10 workers on the graveyard shift A quality control consultant is to select 6 of these workers for indepth interviews Suppose the selection is made in such a way that any particular group of 6 workers has the same chance of being selected as does any other group drawing 6 slips without replacement from among 45 a How many selections result in all 6 workers coming from the day shift What is the probability that all 6 selected workers will be from the day shift b What is the probability that all 6 selected workers will be from the same shift C What is the probability that at least two different shifts will be represented among the selected workers 1 What is the probability that at least one of the shifts will be unrepresented in the sample of workers 241 A mathematics professor wishes to schedule an appointment with each of her eight teaching assistants four men and four women to discuss her calculus course Suppose all possible orderings of appointments are equally likely to be selected a What is the probability that at least one female assistant is among the rst three with whom the professor meets b What is the probability that after the rst ve appointments she has met with all female assistants 251 One box contains six red balls and four green balls and a second box contains seven red balls and three green balls A ball is randomly chosen from the rst box and placed in the second box Then a ball is randomly selected from the second box and placed in the rst box a What is the probability that a red ball is selected from the rst box and a red ball is selected from the second box b At the conclusion of the selection process what is the probability that the numbers of red and green balls in the rst box are identical to the numbers at the beginning Stat 224 Fall 2004 Dec 7 8 2004 Discussion 11 1 Goodness of Fit k 7 I 2 X2 Z 01 61 N 2671 i1 5139 11 Example The discrete uniform distribution The observed data 12345678910 38104310218115 2 Analysis of Variance 21 Example Exercise 1212 page 415 Corrosion rates percent were measured for 4 different metals that were immersed in a highly corrosive solution Aluminum 75 77 76 79 74 77 75 Stainless Steel 74 76 75 78 74 77 75 77 Alloy l 73 74 72 74 70 73 74 71 Alloy ll 71 74 74 73 74 73 71 a Perform an analysis of variance and test for differences due to metals using 04 05 b Give the estimates of corrosion rates for each metal c Find a 95 con dence interval for the differences of mean corrosion rates 1231 MSC ting listatwiscedu Ting Li Lin Stat 224 Fall 2004 Nov 30 Dec 17 2004 Discussion 10 1 Hypotheses Concerning Two Proportions Test statistic 1amp Z 1 2 with p 7X1 1 X2 A A 1 1 711712 M1719 i i 711 712 Large sample con dence interval 1 2 711 712 2 Hypotheses Concerning Several Proportions Test statistic lto e gt2 2 g N XH MN 16 X2 6 H ilj 21 Example Exercise 928 page 307 Textbook A study showed that 64 of 180 persons who saw a photocopying machine advertised during the telecast of a baseball game and 75 of 180 other persons who saw it advertised on a variety show remembered the brand name 2 hours later Use the X2 statistic to test at the 005 level of signi cance whether the difference between the corresponding sample proportions is signi cant 22 Example Exercise 938 page 308 Textbook Verify that the square of the Z statistic on page 304 equals the X2 statistic on page 301 for k 2 1231 MSC ting listatwiscedu Ting Li Lin STAT 2243 Discussion 043007 TA Quoc Tran Single Factor ANOVA 0 One Way ANOVA focuses on a comparison of more than two population or treatment means 0 Null and alternative hypothesis H01M12quot39I 7 Ha At least two of the M s are different where I is the number of populations or treatments being compared J number of observations in each population or treatment n total observa tionsnIJl 0 Basic Assumptions The I population or treatment distributions are all normal with the same a i 0 Goal To separate signal from noise 7 Total sum of square SSTo ELI 211yij 7 32 7 Sum of square due to treatments signal between SST 211 J a 7 y 7 Sum square of errors noise within SSE 211 211yij 7 33 7 SSTo SST SSE 0 ANOVA table J obs in each sample Source df Sum of squares Mean square F Treatments I71 SST MST SSTI71 M71122 Error IJ7l SSE MSESSEIJ7l Total n 7 l SSTo where 7 Test statistic F7value has an F distribution with 11 I71 and 12 J 7 1 when H0 is true and the basic assumptions are satis ed 7 Pvalue PF12 gt F 7 value Reject H0 when Pvalue S a or Fvalue gt critical value EXAMPLES 0 Example 1 In an experiment to investigate the performance of four different brands of spark plugs intended for use on a 125cc twostroke motorcycle ve plugs of each brand were tested and the number of miles at a constant Email tran statiwiscledu 1 RM B248D MSC STAT 2243 Discussion 043007 TA Quoc Tran speed until failure was observed The partial ANOVA table for the data appears below Fill in the missing entries7 state the relevent hypotheses7 and carry out a test Source df Sum of Squares Brand Error Total 310750076 Mean Square F 14771369 0 Example 2 Using Descriptive Statistics Table Problem 197 chapter 9 Treatment 1 2 3 Mean 34 318 322 Standard Deviation 22 13 11 Sample Size 6 6 6 Meaning 7 J6I3n18 7 Treatment mean 1 343 7 Q2 318 I if 2 7 Treatment standard deviation 3 W 222 73 7 Evlfffy2 2 13 Calculating 7 Grand mean g Jy 1 J 2 J3 7 634363186322 7 3 n 7 18 7 7 Sum square of treatments SST ELI J 96 Q 7 Q 6 96 343 7 328 6 318 7 328 6 322 7 328 7 Sum square of errors SSE J 7 Us J 7 08 J 718 5 6 222 5 6 132 5 6 112 Email tran statWiscedu 2 RM B248D7 MSC Stat 224 Handout 4 TA Shubing Wang Oct 5 2005 Summary of Section 31 32 33 1 De nition of random variables functions that map the sampIe space to the reaI Iine A discrete ran dom variabIe is an rv whose possibIe vaIues are accountabIe sequence A continuous random variabIe is an rv whose possibIe vaIues consists of an entire interva on the number Iine Examples 37 b Y the number of students on a cIass Iist for a particuIar course who are absent on the rst day of ciasses So the range of Y 0 1 2 n where n is the number of the students in this cIass And Y is discrete 37 d X the iength of randome seIected rattIesnake So the range of X 5 60 in the units ofinches So we see it is an interva So X is continuous 310 The number of pumps in use at 6pump station and 4pump station a T the totaI num ber of pumps in use So range of T0 1 2 H 10 c U the maximum number of pumps in use at either station Range 0 1 2 3 4 5 6 What about minimum number N De nition of probability mass function pmf of an rv X p7 PX 31 Ps E SXs And the cumulative distribution function cdf of X Fm PX Sm P696 5 I My ltm Examples 312 a Note that the event Y g 50 Using cdf or pdf b Note that the event Y 2 50 c What39s the probabiIity of rst standby person get on the pIace For this case at most 49 of the ticketed passengers must show up PY g 49 05 10 12 14 25For the 3rd person on the standby Iist at most 47 of the ticketed passengers must show up PY g 44 05 10 12 316 a Hint make the tabie x Outcomes pX O FFFF 74 2401 1 FFFSFFSFFSFFSFFF 4733 4116 2 FFSSFSFSSFFSFSSFSFSFSSFF 67232 2646 3 F555 SFSSSSFSSSSF 4733 0756 4 555 34 0081 S b Equot c Look at the Iargest bin in the histogram do PX 3 2 p2 p3 p4 264607560081 3483 This couId aiso be done using the compiement 1PX lt 2 Definition of expected value or mean value of a discrete rv X de ned by EX M 2 mm 16 and that ofa function ofX mm Z M pm 16 The variance of X W mx Em Note that here fX X EX2 A shortcut formuIa for 72 02 War EX2 E002 Ruies of expectation and variance EaX b aEX 1 VaX b tX Examples 731 at Use de nitions EX 1352 1595 1913 EX2 13522 15925 19123 VX shortcutformuia be Use the expectation of a function ct Using ruIes of variance EILX EX 01X2 EX 01EX2 337 Using the expectation of a function EWXH Elt gt 341 at EXX 1 EX2EX EX2 EXX 1 EX a b Shortcut formuia of variance 347 at and b where X Bin6010 ct Either 4 or 5 gobiets must be seIected i SeIect 4 gobiets with zero defects 010 1 014 ii SeIect 4 goblets one of which has a defect and the 5th is good 011 1 013 PX 0 Exercises Practice probiems for midterm 1 7 3 giris 2 guys and 2 identica mannequins are to be arranged in a row at Find the totaI number of arrangements possibier b Find the probabiIity that the 2 boys are NOT together a Note that mannequins are identica so that the number of arrangements is P77 7 2 2 b Using that PNot together 1 Ptogether 1 1102 23 What is the probabiIity of not seeing any 6s in 3 throws of a die Note that the number of cases in sampie space 6 x 6 x 6 So the probabiIity r 5 4 4 5 Not seeing 6 5 63 32 of Homework 302 40 of students in a cIass are civiI engineers and 60 are mechanica engi neersr 20 of the civiI engineers got an A 30 got a B Whereas 15 of the mechanica engineers got an A and 40 got a B at Find the probabiIity that a student who is picked at random from this ciass got an A b Find the probabiIity that the student is a mechanicaI engineer given that heshe got an A c Find the probabiIity that the student is a civiI engineer given that heshe did not get a B d Find the probabiIity that a student did not get a B given that heshe is a civiI engineer er Find the probabiIity that a student who is picked at random from this ciass is a civiI engineer who got an A De ne events Cm from Civi engt Cme from mechanica eng 54 getting A SE getting B 50 getting other scores and we know PC V 5 PC 715 PS 4C 75 PSA61715PSHClt5 PSHC1715 PSOCCB PSOC1715 CL PSA n 1554 PSA n C1715 PSAClt5PCC5 PSACWBPCWB b v 1an 0 SA P1715SA H54 C4 MCCASE PCa505PSs Then caIc WSW 1 PSH Note that PSg can be caIcuIated as PSA and PCamg PSfCCFPCCB PS CCB 1 PSACC5 PSOCC5 PSAOCCB PSACC5PSCC5 duplicate birthdays The first column gives n the number of people randomly selected The second column gives Pall birthdays are different 365 364 363 to n factors divided by 365An The third column gives Psome birthdays coincide 1 Pall birthdays are different You see that there is a better than even chance two birthdays will coincide as soon as you have 23 people and it rapidly increases after that Here is a quick way to find the magic number 23 With a365 you want to solve the equation aa 1a n1aAn 12 for n Divide in the a39s on the left and take logs log1log1 1alog1 2alog1 3a log1 n 1a log2 Expand the logs to first order this is accurate enough as long as a is large 1a 2a 3a n 1a log2 The sum on the left is approximately nA22a yielding n z Sqrta log4 For a 365 this gives n z 2249 which is a pretty good approximation n Pall diff Psome same 1 10000 00000 2 09973 00027 3 09918 00082 4 09836 00164 5 09729 00271 6 09595 00405 7 09438 00562 8 09257 00743 9 09054 00946 10 08831 01169 11 08589 01411 12 08330 01670 13 08056 01944 14 07769 02231 15 07471 02529 16 07164 02836 17 06850 03150 duplicate birthdays OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 6531 6209 5886 5563 5243 4927 4617 4313 4018 3731 3455 3190 2937 2695 2467 2250 2047 1856 1678 1513 1359 1218 1088 0968 0860 0761 0671 0590 0517 0452 0394 0342 0296 0256 0220 0189 0161 0137 0117 0099 0083 0070 0059 0049 0041 0034 0028 0023 0019 0016 0013 0010 0008 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 3469 3791 4114 4437 4757 5073 5383 5687 5982 6269 6545 6810 7063 7305 7533 7750 7953 8144 8322 8487 8641 8782 8912 9032 9140 9239 9329 9410 9483 9548 9606 9658 9704 9744 9780 9811 9839 9863 9883 9901 9917 9930 9941 9951 9959 9966 9972 9977 9981 9984 9987 9990 9992 duplicate birthdays 90 91 92 93 94 95 96 97 98 99 oo OOOOOOOOOOOOOOOOOO 60 50 60 90 40 10 80 80 20 10 OOOOOOOOOOOOOOOOOO H O O O O O O O O O O H 9993 9995 9996 9996 9997 9998 9998 999861 999891 999914 999933 999948 999960 999969 999976 999982 999986 999989 999992 999994 999995 999997 999997 999998 999999 999999 999999 999999 000000 000000 STAT 224 Oct 2122 DISCUSSION 8 Note We say that X 5 7 mp TWO 7 0 is distributed as a standard normal only when X is binomial and np and nl 7 p 2 10 1 The Distribution of the Sample Mean 1 Proposition et X1X2 Xn be a random sample from a distribution with mean value M and standard deviation 0 Then 0 EU Mi M o VX a 072 0 0X In addition with To X1 X ET0 nM VT0 n02 and 0T0 u 2 The Central Limit Theorem CLT 2 Let 1 2 n e a random sample form a distribution with mean M and variance 0 Then if n is 7 2 suf ciently large X has approximately a normal distribution with M2 M and 03927 77 and To also has approximately a normal distribution with MTO nM and 0 n02 The larger the value of n the better the approximation 3 Rule of Thumb If n gt 30 The central Limit Theorem can be used 2 The Distribution of a Linear Combination 1 Proposition Let X1 X2 Xn have mean values M1 M2 Mn respectively and variances of 0 03 072 respectively 0 Whether or not the X s are independent Ea1X1 a2X2 39 39 39 aan a1EX1 a2EX2 39 39 39 anEXn 041141 a2M2 anMn o If X1X2 Xn are independent Va1X1 a2X2 aan a VX1 ailX2 ailX71 7 as 2 EX1 7 X2 EX1 7 EX2 and if X1 and X2 are independent VX1 7 X2 VX1 VX2 3 If X1X2 Xn are independent normally distributed rv7s then any linear combination of the X s also has a normal distribution In particular the difference X1 7 X2 between two independent normally distributed variables is itself normally distributed 3 Point Estimation 0 De nition A point estimate of a paramter 9 is a single number that can be regarded as a sensible value for 9 A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data The selected statistic is called the point estimator of 9 0 De nition A point estimator 3 is said to be an unbiased estimator of 9 if 9 for every possible value of 9 0 Example When X is a binomial random variable with parameters n and p the sample proprotion 5 Xn is an unbiased estimator of p 0 Typically we like unbiased estimators with a preference for unbiased estimators STAT 224 Oct 2122 4 Exercises 550 The breaking strength of a rivet has a mean of 10000 psi and a standard deviation of 500 psi What is the probability that the sample mean for a random sample of 40 rivets is between 9900 and 10200 If the sample size is had been 15 rather than 407 could the probability requested in the previous question be calculated from the given information 564 Suppose your waiting time for a bus in the morning in uniformly distributed on 087 whereas waiting time in the evening is uniformly distributed on 010 independent of morning waiting time a If you take the bus each morning and evening for a week7 what is your total expected waiting time b What is the variance of your total waiting time C What are the expected value and variance of the difference between morning and evening waiting times on a given day 21 What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week STAT 224 DISCUSSION 7 1 Continuous Random Variables and Probability Density Functions H Continuous Random Variable A random variable X is said to be continuous if its set of possible values is an entire interval of numbersthat is7 if for some A lt B7 any number 1 between A and B is possible 2 Probability Density Function Let X be a continuous rvi Then a probability density function pdf of X is a function such that for any two numbers a and b with a S 127 b Pa S X S b 3 Cumulative Distribution Function Fltzgt PltX z MW 0 PXgtaliFa o PaSXSbFb7Fa F I 1W 4 Expected Value and Variance for Continuous random variables o m EltXgt mom mm o 0 vltXgt Egon e m2 mm o W M 7 E0012 2 Uniform Distribution A continuous rv X is said to have a uniform distribution on the interval A7 B if the pdf of X is L fltzABgt 1 A3133 0 otherwise EX L33 VX Lg z 3 Normal Distribution 1 2 1 57 20M 27f Ur fr o Mean7 and Standard deviationai We say Z has standard normal distribution7 if Z has normal distribution with mean M0 and standard deviation 01i X in U o If X has normal distribution with mean M and standard deviation 0 then Z has standard normal distribution 4 Exponential Distribution o It is frequently used for the distribution of times between the occurrence of successive events 0 fze 120 Fzlie 7 7 120 o EX VX 71 Integration by Parts STAT 224 5 Exercises 414 The article Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants77 Water Re search7 1984 11691174 suggests the uniform distribution on the interval 7520 as a model for depth cm of the bioturbation layer in sediment in a certain region a What are the mean and variance of depth b What is the cdf of depth C What is the probability that observed depth is at most 10 Between 10 and 15 d What is the probability that the observed depth is within 1 standard deviation of the mean value Within 2 standard deviations 437 The automatic opening device of a military cargo parachute has been designed to open when the parachute is 200m above the ground Suppose opening altitude actually has a normal distribution with mean value 200m and standard deviation 30m Equipment damage will occur if the parachute opens at an altitude of less that 100m What is the probability that there is equipment damage to the payload of at least one of ve independently dropped parachutes 442 The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 70 and standard deviation 3 Rockwell hardness is measured on a continuous scale a If a specimen is acceptable only if its hardness is between 67 and 757 what is the probability that a randomly c osen specimen has an acceptable hardness b If the acceptable range of hardness is 70 7 c 70 c7 for what value of 5 would 95 of all specimens have acceptable hardness C If the acceptable range is as in part a and the hardness of each of ten randomly selected specimens is independently determined7 what is the expected number of acceptable specimens among the ten 21 What7s the probability that at most eight often independently selected specimens have a hardness of less than 7384 459 Let X denote the distance m that an animal moves from its birth site to the rst territorial vacancy it encounters Suppose that for bannertailed kangaroo rats7 X has an exponential distribution with parameter A 01386 as suggested in the article Competition and Dispersal from Multiple Nests77 Eculugy7 1997 873883 a What is the probability that the distance is at most 100 m At most 200 m Between 100 and 200 m b What is the probability that distance exceeds the mean distance by more than 2 standard deviations C What is the value of the median distance
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