Week5-4up - 95 STA 2023 c B.Presnell & D.Wackerly -...

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Unformatted text preview: 95 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 96 Last Time :  § £ ©¤¦¥¨¢ and . What is §£ ©¨¦¥¤¢ ¡  HIV Example (Like #3.109, p. 158) can count; and those who can’t. A  C  %5¦¥8@7%5¦¥43¥  © # £ B6  © # £ 9 ! £¤¢201)©¤¦¥¨("%$¦¥¨¢  ©  § £¢ !© # £ '  &¥¤¢ £  © # £ ! § ©£ %$¦¥¨¢ " ¥ ¨¢  Assignments :  § ©£ ¥ ¤¢ Know Thought: There are three kinds of people: those who (F. S.) For Today: pages 172–176 For Tuesday: (F. S.) mutually exclusive Exercises 4.22, 4.27-29   ¤21  ©¨¦¥¤("  %$¦¥¨¢  ©£ ¢ 0 § £ ¢ ! © # £   $&¥¤GF%$¦¥¨(D¦¥¨¢  © # £ ¢ E © # £ ¢ ! £ QUIZ #2 Covers Chapter 3 and Project 1 Read pages 179–185  For Wednesday: (F. S.) Put the pieces together! ¡ Discrete and Continuous Random Variables ¡ Probability Distribution for Discrete Ran. Var. For Thursday: (p. 166). Exercises 4.33, 4.36 97 (p. 169) STA 2023 c B.Presnell & D.Wackerly - Lecture 8 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 98 EXAMPLE : Have 3 light bulbs, 2 good, 1 defective. # defective. ”Label” the bulbs discrete r.v. is a formula, table, or graph giving : Each possible value for the variable ¡ the prob., , associated with each possible value of the r.v. Note: Must have: for all poss. values ; . H ! £ v t "s¦Hgf w¡ t £ rp ur s¦Hgf q¡ second. £ h¦Hif first, then W means select RECALL: (p. 169) The probability distribution of a ¡ PI WTVUITSPQI RR ! H Randomly select 2, ? UI `W SI BYI WU WP P I P IU U IP Y`W SI BYI £ h&Hgf H ¡ ¡ WP I ¡ !X ' ! b £ ¡ Pe Ddc&a¤¢ ¡ 99 Ex. Raquetball: odds are 2:1 that when H and T play, H by H. ! H will win. H and T play 3 matches. # matches won STA 2023 c B.Presnell & D.Wackerly - Lecture 8 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 100 The Expected Value a Random Variable Consider the lightbulb example. Repeat experiment a Sample roughly about 9 CU   over 600 repetitions should be £ p A t p t !  ¤¡ t £ 0  ¤¡ t £ 0  ¤¡ t £ £ £ £ £ s¦Hif ! H ! ! £f " t g¤0  t £ E  p g¤0  p 4! £f £ 101 times £0 t E t£ 0 p pp E pp  p p #0 t p p 0 p pp  E  t E  !E t E p V! E @ BC A @ 9 t £ STA 2023 c B.Presnell & D.Wackerly - Lecture 8 times;   " So, average value of £ TTT t £ TTH £¤¢ ¡ £¤¢ ¡ £¡ ¤¢ THT roughly H !  ¤¡ £ 0  ¤¡ £ 0  £ £ !  ¤¡¢ £ 0  ¤¢t £ 0  £ £¡ !  ¤¢t £ 0  ¤¥ £ 0  £¡ £¡ THH ©¢¨¡ ©¢¨¢ ¡ ©¢¨¢ ¡  ©¢¨¡ ©¢¨¢ ¡   ©¢¨¡   ©¢¨¡ ¦ § © ¢¨¡ HTT ! t H ! p H HTH H HHT £ HHH large number of times, say 600. Expect ! p p §" p p £ U p p !  p p £ P Probability Point STA 2023 c B.Presnell & D.Wackerly - Lecture 8 EXAMPLE : Suppose # of calls to an office in a 5 min. period has distribution Def. (Def. 4.5, p. 172) For a discrete r.v. , the H H H £ s¦Hif H v Ds¦H`a ! ! £ is NOT necessarily a possible value of . ! H ! ! s¦H`a ¡ £ H £ s¦H`a ¡ of .  h¦H£ a ! $ is a “weighted average” of the possible values $ defined to be (population mean of ) is ' t' £ %' t £' tp £ s¦Hif H expected value of 102 103 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 104 Def. (Def. 4.6, p. 174) The variance of a discrete r.v., , H EXAMPLE : #4.27 p. 177 Rock concert: is No rain, profit of $20,000, Rain, lose $12,000 profit. Recall that £ h¦Hif H £ ¨¢ The standard deviation of . (Def. 4.7, p. 174) is H ! %H  ' p " ! . '  H ¢  ¨ U h¦H£ P P ©§ ! U ¦ £ h&Hgf U  $ £&H£ v ! ¥U  $ £¦H£ ¡ a ! U ¢ ¤¢ rain  U ! ! £ Ds¦H`a ! A shortcut formula for the variance of is ¢ £a U $ ¢s¦H£if U H v ! U $  U ¦H`(! U H ! 105 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 EXAMPLE : Telephone Calls is and  !  Interpretation of H The standard deviation of . The population mean and standard deviation ¡ Tchebysheff’s Theorem (always applies) ¡ Empirical Rule (if mound shaped) If we have the probability distribution, we can “do better” than ¡ ' t' £ '% t £' tp £ h¦Hif H ! ¤¢ ¥U  $ £&H£ ¡ a ! U '! t$  ! – The “at least” – Tchebysheff. ! – The “approximately” – Empirical Rule 106 ! 107 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 108 Ex. According to 1990 Statistical Abstract of the U.S., of 455 million revenue air passengers 1988, there were 285 fatalities due to air accidents. Flight insurance ! " R R ¦H¨¢ !£ £ tp ! " ' &H¤¢ £ p ' ' £ ! ' ' ¤ £ R p ¡ " ¦ £  t ! $  ! " !£ t ¦H¨¢ ' ! "  ' ¦H¤¢ £ ¦ RRR H  ' £ ' t ' p  R¦ ! ¦' ' t ! $  '' '¤ p¡ p ! R  t ! $ ¡ from a single policy. . Estimate prob. of fatal crash to be and ¦ %¡ ¢% ¢% or between be the “profit” to the insurance company £ h¦Hif H ' p p p p p p ¢p p p R p p p R ¡ are crash. Let  between ¦ Possible values of costing $10 will pay $1 million if the passenger dies in a H EXAMPLE : Telephone calls . and or ! "h¦H£ a ! 109 STA 2023 c B.Presnell & D.Wackerly - Lecture 9 STA 2023 c B.Presnell & D.Wackerly - Lecture 9 110 Last Time : ¡ ¡ Variance p. 174 ¡ The standard deviation of $ us just don’t have film. Today : Pages 179 – 185 H U  ! £ s¦Hif U  $ ¢ ¦H£ v ! ¥U  $ £&H£ ¡ a ! U ¤¢ Assignments : For Thursday : £ P ¡ no sale – probability is $50,000 sale – probability is ¡ Find the distribution of ! H EXAM 1 PP Each contact results in , resp.  and  For Tuesday : day. Probabilities are P For Monday : OPTIONAL REVIEW (p. 174) is EXAMPLE : Salesman contacts 1 or 2 customers per U Exercises 4.33, 4.36 £ s¦Hif H v "s¦H`a ! ! £ Thought: We all have photographic memories–some of Mean p. 172 daily sales. . 111 STA 2023 c B.Presnell & D.Wackerly - Lecture 9 113 STA 2023 c B.Presnell & D.Wackerly - Lecture 9 H A sample point must indicate: 112 £ h¦Hif STA 2023 c B.Presnell & D.Wackerly - Lecture 9 ¡ How many contacts were made. ¡ Sale(s) made. 1 contact, sale $ Sales ! ! £ U $ ¢ h¦Hif U H v ! U R R        ¢   ' ¤ ¦¦ ' £ E p £' ¦ £ ¢£ £ £ ¢£ £ £ £ R '  p t t t R ¢% % R p %  ¢  R ¥ ¢ ¤ R ¦¦ R E p¡ ! ¦ £   Ue P !  £ P    !  P ¡ ¢P £ P ! P P    ¡ ¢ £ X P 0  £U P 0  PU P 0 U  X "X X £ t ! STA 2023 c B.Presnell & D.Wackerly - Lecture 9 ¤© p pppR p% pppR p% pppR ppt Xt P !  P P 0 P pppR p% ! ! Probability ! X Xt ! Sample Point  h¦H£ a ! 2 contacts, sale on 1st, no on 2nd 114 Binomial Experiment p. 179 identical trials in experiment. § 1. 2. Each trial results in one of two possible outcomes, . stays same from trial to trial. Y¨£¨¢(! f ¢ t ! 3. Prob. of X X £  X ¤(! f £¢ © '' H ! t f ! p t w§ !X   , ¨ # H in 10 tosses, , ! Ex. Toss a fair coin 10 times,   trials. ’s in the . ¨¡ ! 5. Variable of interest is , the number of is in this interval ? . ! H Ex. Choose 10 microwave ovens at random from X factory output. Count number defective ( =defective, f =good). unknown. § 4. Trials are independent. X ! $ p R¤t ! ¡ ©% ¦ ¨ H ¡ What is the probability that or ¨ Thus, for the sales example, ¡ 116 Fact: The number of ways of selecting =Henry wins, from among objects , where the order in which the § X =Thomas wins ¡ Ex. Play three games of racquetball STA 2023 c B.Presnell & D.Wackerly - Lecture 9 ¡ 115 STA 2023 c B.Presnell & D.Wackerly - Lecture 9 ¨ objects are selected does not matter is '  ¥¢ § £ 0 ¡ ¡ § ¦ §¡ § ! ¡  £ ¤¡ ¢ § £ ¤¡ p!  ¢ §  £ ¤¡ H ¡ . ! £ ¡ ! ¡ t! ¢ p St  size £ H § § t0 § ! 0 1 ¢ § £ 0  t ¢ § £ 0  !  ! §  ! £ t : ¢  size pt ! STA 2023 c B.Presnell & D.Wackerly - Lecture 9 118 § Note ¡ Each single sample point where corresponding to . probability . Thus Try the formula !£ " if  ! w§  ! " £if ! £ H ¡ £¢    £ t t ¨ X "X X "X X © t Thus, has . ! H R ¡ Probabilities . !£ " gf ! £ "s¦Hgf ¡ £ ¨H – toss 3 coins, look at 3 microwaves, etc. p U©f U©f U©f f0f0 U f ! Qq¤Y©  © ! w§ Sample Points distinct sample points R''' ¢ § Why this formula? Sample point approach: § for There are ! %H . R t R p H © ©§ © f ! ¨© R'''R RtR p H are § Possible values of ! H  § ¡ £¢ X ¨(! H f . (p. 183) ¡ trials and ¡ !  ¡ ! % ¡ t ! p  ! 117 is a binomial random variable with X"X ¨ X¨X ¨ X ¨ ¨ X ¨ ¨ ¨ X ¨ ¨ ¨ Ex. : . committee of factorial is STA 2023 c B.Presnell & D.Wackerly - Lecture 9 If , read ¡ is a positive integer, if choose . EX. : Club consists of 10 members, select a Some Preliminary Results Def. : (p. 139) If is read if § ¢ Variable, . –The Probability distribution of . § associated with all values of a Binomial Random § Objective : Obtain a formula for the probabilities Ex. Political Poll : Likes Bush, Not r Ex. Diagnostic device : detects or misses Same as sample point approach!! 119 Ex. H and T play 3 matches of racquetball. H has prob. 2/3 of winning any match. Matches indep. ¨ § H ¢ ways to do this, so there are sample points with X £ sample points corresponding to the value . !£ D if ! ' !"h&Hgf £ H Thus ! %H ’s. , # won by H. , £ ¡ – There are ! w§ , and then filling in the rest as £ § C ¨ !   A!@ X ¨ X 9   positions to be filled ! H with an of the £¢ ¡ – Think of selecting !f § positions £¡ ¤¢ H¢ ¨ – One sample point : ! H ¡ X How many such sample points are there? t H . ©¢¨¡ © ¡ ¢¨¢ H © has prob. © ¡ ¢¨¢  !  £ 0  ¤¥ £ 1 ¤¡ t £ 0£ ©¢¨¡ £¡ ¤¢ £¡ © ¢¨¡¢ !  £ 0  ¤¡ t £ 1 ¤¢ £ 0 £¡ © ¢¨¡ £ ¤¡¢ £ £ 0  £ 1 £ 0 ¨ so each sample point corresponding to the value £¡ ¤¢ ’s in some order, HHH HHT HTH HTT THH THT TTH TTT £¡ ¨¢ successes each have Probability ! Outcome ¦ § in £¡ ¤¢ ’s and . 120 © ¢¨¡ Sample points that give for each f the sample point, and a for each X prob. is calculated by multiplying a STA 2023 c B.Presnell & D.Wackerly - Lecture 9 H ¡ For any sample point of a binomial experiment, STA 2023 c B.Presnell & D.Wackerly - Lecture 9 3 2 1 2 1 1 0 There are . Each has probability ¡ ...
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This note was uploaded on 12/15/2011 for the course STA 2023 taught by Professor Ripol during the Spring '08 term at University of Florida.

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