Week4-4up_001 - 74 STA 2023 c D.Wackerly - Lecture 6 STA...

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Unformatted text preview: 74 STA 2023 c D.Wackerly - Lecture 6 STA 2023 c D.Wackerly - Lecture 6 75 The trouble with jogging is that, by the time you realize you’re not in shape for it, it’s too far to walk back. Assignments Conditional Probability For Today : p. 121–124, 127–135 7 5§ £ 86%1#¤¢ is # (Quiz # 2), similar graphs (different data set)–similar . The conditional probability of § #£ %A'¢ §# 4 £ ©§ 9 £ %$"¦¥'¢ %@#¤¦¥'¢ PROJECT 1 : NOT COLLECTED!!! BUT–on next quiz be two events, where questions will be asked about the INTERPRETATION of given B For Wednesday : p. 127–135, Read pages 163 – 170 and ¥ Def. (p. 122) Let ¥ 3.47, 3.49, 3.50, 3.52, 3.53, 3.55–3.57, 3.59, 3.60, 3.61 # For tomorrow: Probs 3.35, 3.37–3.39, 3.41, 3.43, 3.46 the graphs. # LAST TIME : ¥ (What proportion of ¡ ¡ Additive Rule of Probability p. 117 #  ¦¥¤¨¦¥¤¢ § £ ¢   © § £ Complement p. 115 : is also in ?) ¥ ¡ Mutually exclusive (disjoint) p. 118 § # 4 £ ¢ § £ ¢ )§ £ ¢ ©§ # ! £ %$"(¥'321#'"0(¥'&%$"¦¥¤¢ 76 STA 2023 c D.Wackerly - Lecture 6 STA 2023 c D.Wackerly - Lecture 6 77 EXAMPLE : Pol. party – job type. ¡ If I know that a person is a Republican, what is the  hY probability that (s)he is a white collar worker? ¥ RW DW IW RB DB IB ¡ ¡ ¡ # ¡ ¡ ¡ . Thus, §£ %1#¤ © %$"¦¥¢ ¤¢ %‰#'¦¥¤¢ §# 4 £ ©§ 9 £ Q  © § # y£ hˆSvi%‡A'¢ †…Hi%1t¤¢ Q ©§# £ H ©§ u £ xvi€1y'¢ xwviv¨1t'¢ H ©§ u £ ¡ get a number greater than 3 S ©§#£ †…7v%P¦‚¤¢ W ©§ u£ „ƒ7viR¦‚¤¢ ¡ get an even number person is a Republican . B §£ 21#' © © §2ps¦¥¢ '¢ %@#¤¦¥'¢ # 4 £ ©§ 9 £ ©§# 4 £ %$"¦¥'¢ Y C©#4 rqp"¥ ©§ £ %1#'¢  hY C D© Y C© 8# ©§ £ i¨¦¥'¢ CD©gY C© f&¥ © §c £ ¢ ed(b¤a`  Y WF F SF QF HF C EEXPV UTRIPIGED© B . person is a white collar worker . Y C8# ¡ © C© f&¥ ¡ Balanced Y EXAMPLE : Toss a balanced die. If a person is NOT a Democrat, what is the probability that (s)he is a blue collar worker? ¡ 78 STA 2023 c D.Wackerly - Lecture 6 STA 2023 c D.Wackerly - Lecture 6 EXAMPLE 3.13: (p. 123) develops cancer © q# Simple Events Probabilities #  $4  ¥ #$4  ¥  ps¥ #4 #4 $"¥ Q Gƒ7 7 †…H V „ƒ7 Why? Recall (Defn of Cond‘l Prob.): §£ (¥' F %$"¦¥¢ ¤¢ ¨¥ p1#¤¢ §# 4 £ ©§ 9 £ H so §£ ¨¦¥¤ © %$"¦¥¢ '¢ ¦¥¤&¥ p1#¤¢ ¦¥¤¢ §# 4 £ § £¢ ©§ 9 £ § £ ©¥ Given: Multiplicative Law of Prob. (p. 128) §%@#¤¦¥¤¢ %1#¤&© 9 £ § £¢ § 9 £ § £¢ ©§# 4 £ ¥ $1#'¢ ¨¦¥¤&%ps¦¥¤¢ individual smokes 79 ©i§  ¦¥¤¢ £ ©§ £ i¨¦¥'¢ § £  ¦¥¤¢ © Find the probability that a person develops cancer Useful for computing the probability of the intersection given that the person smokes. when : © © §£ ¨¦¥'¢ §£ ¦¥¤¢ §#4 £ 2$"¦¥'¢ © €&1#'¢ ¥ p1#¤¢ §¥4 £ ©§ 9 £ ¡ Conditional Probability is given ¡ 80 STA 2023 c D.Wackerly - Lecture 6 Conditional Probability easy to get without definition. STA 2023 c D.Wackerly - Lecture 6 81 EXAMPLE : (Screening for quality) Shipment of 30 cameras, 6 defective. Choose 2 at random. If both EXAMPLE : (#3.61, p. 137) Microchip: good, accept shipment. Find the prob. accept shipment. © R7 v§ camera good  © ¥ ¨¦ ©§ . Find the probability that a microchip does not fail : If first camera known to be good, what is (cond.) prob. during the first year. NOTE: If chip does not fail during first year–It cannot fail second is good? on its first use!!! H© &#   Y when we select the second, . cameras, Note that good, shipment accepted ©§ 9 £ § £¢ © §# 4 £ i¨¥ p1#¤¢ ¨¦¥'&&2ps¦¥¤¢ Y © §¨¥¦'¢ ¥ p1#¤&%$"¦¥'¢ ¡ £ § 9 £¢ ©§# 4 £ © ¡ “Does not fail during first year” there are bad. Y Chip lasts from first use through first year , CD©q#$"¥ 4 ©§ 9 £ ¥ p1#'¢ ¡ . : Start with 30 cameras, 6 bad. If first chosen is good, ¡ C© 8# ¡ ©i¨¦¥'¢ §£ C© f&¥ ¡ Microchip lasts through first use , camera good ©§ 9 £ ¥ $1#'¢  ¡¡ £¢ is  £ '¢ If it lasts through its first use, prob. that it lasts a year ©§ £ ¦¥¤¢ fail on first use . ¤ ¥ 82 STA 2023 c D.Wackerly - Lecture 6 83 §£ %1#¤ © %$"¦¥¢ ¤¢ i2@#'¦¥¤¢ §# 4 £ ©§ 9 £ Independence Does knowing that one event has occured alway have an impact on the probability of another event? © 9£ i§ ¢ '¦¥¤¢ DEFINITION: (Def. 3.9, p. 131) Events said to be independent if  ¡ § c (b'¢ © £ © 4 £¢  § ¢ "(¥'aY C D© ¢ "¥ ¡ 4 ©i2$"¦¥£'aY §#4 ¢  C©#4 rqp"¥ ¡ ©§ £ ¤ ©§ £ ¡¨©© § ¢ £'¢ ¤¡§i%#1¤¢ ¦ © ¡¥i¨¦¥'¢ ¡  Y W F SF C EEXFPV UTRQ D©Y C¢ 8© £¡ YW V C hEIFPIF S D© Y C© 8# ¡ hEIUTRH CDgY Y WF SF © C© f&¥ ¡ Y WF F SF QF HF C EXPV UTRIPIGED© ¡ “Not independent” = “DEPENDENT.” Ex. Die Toss: ¥ and and , © §¥ 9p1#£'T§¦¥¤&%$"¦¥¤¢ ¢ £¢ ©§# 4 £ # ¥ Note that if STA 2023 c D.Wackerly - Lecture 6 84 and are independent then [by Mult. Law] [by independence] , and ¥ # greater than 2 # # greater than 3 ¢ B get an even number or § £¢ ©§ 9 £ ¦¥¤&i2@#'¦¥¤¢ and are  21#'&¨¥ p1#¤¢ § £¢ ©§ 9 £ STA 2023 c D.Wackerly - Lecture 6 85 Ex. (LAST TIME!!!) Odds are 2:1 that when H and T This is another way to define independence: (p. 133)    21#'¢ ¨¦¥¤&%# 4 ¦¥'¢ § £ § £¢ ©§ £ and winner of each recorded. Odds are 2:1 that H wins means H wins with prob. WARNING Q H i§  d1HUH  © ) £ Independent play racquetball, H will win. H and T play 3 matches, . T wins with prob. Q   ¡ EXAMPLE : Toss a balanced die. and ¥ ? # § £¢ ©§ 9 £ ¨¦¥'&%@#¤¦¥¤¢ Is STA 2023 c D.Wackerly - Lecture 6 . Probs. given Do NOT confuse INDEPENDENT with MUTUALLY in the example for 8 possible outcomes were calculated EXCLUSIVE. assuming matches independent : HH §  £ §  §  †&© hQ 1H hQ T£ hQ $ £ ¤  ¤ ¤ ¤ TTT ¤ TTH ¤ THT ¥ 7 %1#¤¢ wvi¨¦¥'¢ © § £  © § £  # ¥ ©§ 9 £ 2@#'(¥'¢ §21#' £ © §%$"¦¥¢ £¤¢ ©§%‰#9'¥¦¤¢ £ #4 # are THH ¤ and HTT ¤ are mutually exclusive HTH H†#S © hQ 1H hQ A£ hQ $ £  §  £ § H §  H†&© hQ   hQ T£ hQ  1H£ H §  £ §  § H §  £ §  § †#S © hQ 1H hQ T£ hQ  1H£ and that and , Suppose that ¤ HHT ¡ ¡ Independent – need to check PROBABILITIES. ¤ HHH Probability H ! §  £ § H § † "© hQ 1H hQ A£ hQ  1H£  common. Outcome sets have no points in Mutually Exclusive ¡ 86 STA 2023 c D.Wackerly - Lecture 6 STA 2023 c D.Wackerly - Lecture 6 87 EXAMPLE : Space shuttle Challenger disaster. The all six field joints (sealed by O-rings) work. first field joint works C D© ¡ u sixth field joint works . . . . ¤ ¥ (independence) Assuming joint failures indep.: Y © © § #£ § 9 £ ¢ © § # 4 £ %A'¢ %‰#'¦¥¤&%$"¦¥¤¢ §#! £ %$"¦¥¤¢ %@#¤¦¥¤IF 2p"(¥'¢ §9 £¢§#4 £ Q H ©§ £ ©§ £  %1#¤¢  ¦¥¤¢ and Rogers commission concluded disaster results unless C D© u Find are Y ¥ independent events: and # EXAMPLE : Like #3.52(a), p. 136  UQ ¦ ¤¢   § W¡ £   © Uƒ7v© h! „  f§ £  ! „ vi§ ¡ Au¤&P¥¤£§ Au¤¢ ¡ GW W  © £¡ ¢ © ¢ ¢ ¢ © £ 7 © § £ ¤¢  © © § £ ¤¢  © u£ © ¢¢¢ © u£ v§ ¡ e'¢ P¥¤£§ A¤¢  ¡ UQ § ¡ A¤¢ § A'T§ ¨ A'¢ § ¤ A¤¢ § ¦ A'¢ § A'&© u£ u£ ¢ u£ u£ u£ u£ ¢ § ¡ u 4 u 4 ¨ u 4 ¤ $4 ¦ $4 e'&§ u u u£ ¢ © £ ¤¢ all six work Under Challenger flight conditions ( ©§ 9 £ %@#¤¦¥¤¢ (independence) F), estimate £ , so ¡ all six work at least 1 fails , © at least 1 fails 88 F. STA 2023 c D.Wackerly - Lecture 7 ¡¡ £¢ -th of what it was at about , only ¡ Note: At ¡¡ £¢ ©§# ! £ %$"¦¥¤¢ (Add. Law) STA 2023 c D.Wackerly - Lecture 7 89 Thought: Whether you think you can or you think you EXAMPLE : Space shuttle Challenger disaster. The can’t, you’re right! Rogers commission concluded disaster results unless all six field joints (sealed by O-rings) work. C D© u first field joint works C D© ¡ u sixth field joint works . . . For tomorrow: Exercises 3.99, 3.100, 3.105-109, 3.114, 4.3–4.5, 4.7, 4.11, 4.14, 4.16 For Monday: Read pages 172–176 Assuming joint failures indep.: QUIZ 2 – Chapter 3 – Project 1  UQ ¦ ¤¢   § W¡ £   © Uƒ7v© h! „  f§ £  ! „ vi§ ¡ Au¤&P¥¤£§ Au¤¢ ¡ GW W  © £¡ ¢ © ¢ ¢ ¢ © £ 7 © § £ ¤¢  © ! v©  £ © £ ¤¢ ¡ §  q©      v§  © u£ © ¢¢¢ © u£ v§ ¡ e'¢ P¥¤£§ A¤¢  ¡ UQ § ¡ A¤¢ § A'T§ ¨ A'¢ § ¤ A¤¢ § ¦ A'¢ § A'&© u£ u£ ¢ u£ u£ u£ u£ ¢ § ¡ u 4 u 4 ¨ u 4 ¤ $4 ¦ $4 e'&§ u u u£ ¢ © £ ¤¢ For Tuesday:Exercises 4.22, 4.27-29, all six work Under Challenger flight conditions ( £ , so ¡ ¢ ¡ ¢ £ ¡ ¢ ¤ £ ¡ ¢ ¤ ¡ ¢ all six work  at least 1 fails , about -th of what it was at ¡¡ £¢ at least 1 fails ¡ Note: At ¡¡ £¢  %A'¢ ¦¥¤&%$"¦¥¤¢ § § #£ § £ ¢ © § # 4 £ § £¢ ©§ 9 £ %1#¤&¥ p1#¤¢ § § £¢ ©§ 9 £ ¦¥¤&%@#¤¦¥¤¢ § # ¥¡  ¨¦¥¤¢ ¥ $1#'&%1#¤¢ %@#¤¦¥¤&i%i¨¦¥'¢ § £ § 9 £¢ ©§ £ § 9 £¢ ©§#4 £ ¤ ¤ Independent (p. 131, 133): F), estimate £ ¡ Multiplicative Rule (p. 128): F. , only Last Time : ¤ and Y Today : p. 163–169 Y Assignments : 90 STA 2023 c D.Wackerly - Lecture 7 STA 2023 c D.Wackerly - Lecture 7 EXAMPLE : (Like # 3.109, p. 158) : Telephone test for Altzheimer’s disease. 91 A person age 65-69 is randomly selected and tests : person tests positive positive. What is the (cond.) prob. that the person : person tests negative actually has Altzheimers ? : person HAS Altzheimers : person does not have Altzheimers § £ £ §£ €@y9 '¢ § ‡1y¤¢ €1y'¢ ¡ ¡ ¡ ¡ Since  ‡y y ¡¢H„! v§  ‰9  ¤¢  © y £ 7 ©§ £ G! v‡‰y9 '¢ § 65-69, 7 in 1000 have Altzheimers, then for person age : © 4£ i§ v1y¤¢ ¡ B §  1y¤a © £¢ © ©§ £ i€1y¤¢ ©  ‡y 92 y STA 2023 c D.Wackerly - Lecture 7 STA 2023 c D.Wackerly - Lecture 7 93 4 y  § 4 ‡1y£ !0§ 41y£q© § ¤¢ £ ? are mutually exclusive § § 9 1y¤¢ £ , found . “Good” test? according to rules of chance ( the “random” part) Two Types of Random Variables DISCRETE (Def. 4.2, p. 166)– the number of different values of the variable is countable ( look for variables where the values change in “jumps”). – Number of accidents in G-ville next week. – Number of phone calls to fire dept. in a day. – last digit in social security number. – Toss a coin, bet $1 on “heads”, Payoff   %)  4 y QGƒ7 £ © § '¢ £ y£ § v1y¤¢ § 9 A'¢ 4 £ © § 9 1y¤¢ £ SQS  †„†! v© S ! wf) †„„ƒ7v©  WV7  ) †„„ƒ7v© WV7  ) © § 4  1y¤") § v1y¤&i§ ¤¢ £¢ 4 £¢© £ ? takes on different values (the “variable” part) ¡ ¡ ¡ How about Intuitively : a random variable is a quantity that ¡ § Thus and Chapter 4 : Discrete Random Variables ¡ § Note that §£ €@y9 '¢ KNEW . © £ § ‡@y9 ¤¢ H © „! v§  @y9  ¤¢ £ 7 ©§ £ G! i€@y9 '¢ § '¢ £ £ § v1y¤¢ i§ 9 1y¤¢ 4 £ © “specificity of test” According to the Framingham study, for those age How about , and “sensitivity of test” 65-69 selected at random , Know G-ville Sun, 11/12/01, “good test” because Want , § ‡@y9  ¤¢ £ Know or . ¡ 94 STA 2023 c D.Wackerly - Lecture 7 STA 2023 c D.Wackerly - Lecture 7 CONTINUOUS (Def. 4.3, p. 166) – the variable can 95 take on all values on a line interval (look for ¡ variables where the possible values change smoothly, without jumps) – Time to assemble automobile DEFINITION : (Def. 4.4, p. 169) The probability – Height, Weight , Blood Pressure distribution of a discrete r.v. is a formula, table, or – Amount of rainfall (in inches) in a month graph giving : ¡ Each possible value for the variable ¡ that assigns one and only one numerical value to each sample point. Ex. Roll two dice. 36 sample points. Let value of the r.v. for all poss. values ; © § £ ¡ §¡ ¦ ¢ ¡¢ D¥§ £ ¤£7 ¡ © §HF „I†T£ §  †T£ F . . . , associated with each possible Note: Must have (grey shaded box, p. 169): sum of up faces. Sample Point the prob., § 3¡ £ Def. A RANDOM VARIABLE (Def. 4.1, p. 164) is a rule (sum is over all values of ). . . . § WF hXR1W£ 96 STA 2023 c D.Wackerly - Lecture 7 STA 2023 c D.Wackerly - Lecture 7 97 Ex. Shipment of 30 cameras, 6 defective. Select 2 at # good selected (0, 1, or 2). : 1st camera selected is good : 2nd camera selected is good § £ ¡ © §  9¥  1#'¢ §  ¦¥¤&§  $4  ¦¥¤&„&© ¤&h(73¡ £ £¢© # £ ¢ © §H £ ¢ © § £ # ¥ © random. Table of the dist. of : 0 1 2 © § ¨ 9¥  1#'¢ ¨¦¥¤¢ ) §  ¥ p1#¤¢ §  ¦¥¤&© £§£ 9£ £¢ §  ps¦¥'"02p4  ¦¥¤&© # 4 £ ¢ )§ # £¢ Y # 4 !§ # X§  $s¦¥£ 0%$4  ¦¥PC &§ © ¤&§  3¡ £ ¢©  £¢© £ © § 9 £ § £ ¢ © § # 4 £ ¢ © §H £ ¢ © § £ ¥ p1#¤¢ ¦¥¤&%$"¦¥¤&„&© ¤&„1H3¡ 98 STA 2023 c D.Wackerly - Lecture 7 STA 2023 c D.Wackerly - Lecture 7 99 Ex. Sum of up faces on two dice. (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) Distribution of the Sum of Two Fair Dice p(x) 2 (1,1) § £ ¡ 3 (1,2), (2,1) 2/36 4/36 4 (1,3), (2,2), (3,1) 3/36 3/36 5 ... 4/36 2/36 6 ... 5/36 1/36 7 ... 6/36 0 8 ... 5/36 9 ... 4/36 10 ... 3/36 11 (5,6), (6,5) 2/36 12 (6,6) 1/36 Sample Points STA 2023 c D.Wackerly - Lecture 7 © # defective. yF¦ F first, then second. ¦ y y¦ y ¦ means select y y ¦ § £ ¡ ¡ ”Label” the bulbs 5/36 x 1 100 EXAMPLE : Have 3 light bulbs, 2 good, 1 defective. Randomly select 2, 6/36 1/36 2 3 4 5 6 7 8 9 10 11 12 © B y ¡ ¡ ¡  ¡ ied(b'¢ ¡ © §c £ ...
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