Week4 - STA 2023 c B.Presnell& D.Wackerly Lecture 6 72...

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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 6 72 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 For Today : p. 127–135 For tomorrow: Probs : 3.47, 3.49, 3.50, 3.52, 3.53, 3.55–3.57, 3.59, 3.60, 3.61 For Wednesday : p. 127–135, Read pages 163 – 170 PROJECT 1 : NOT COLLECTED!!! BUT–on next quiz, similar graphs (different data set)–similar questions will be asked about the INTERPRETATION of the graphs. LAST TIME : ¤ ¥£ ¢ ¡ Complement p. 115 : ¦  ¡ Mutually exclusive (disjoint) p. 118 © ¨ § ¦ ¤ ¥£ ¢ ¡   ¤ ¥£ ¢ ©¦  £ ¢  ¦ ¤ ¥£ ¢ § ¦   ¤ ¥£ ¢  Conditional probability p. 122 ¦ Additive Rule of Probability p. 117 ¦  ¦  ¤ ¥£  £ ¢ ¢ § ¦  ! ¤ ¥£ ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 73 EXAMPLE : Pol. party – job type. person is a white collar worker . ¡ person is a Republican . ¡ § ¡ ¤ § ¡  If I know that a person is a Republican, what is the ¡ probability that (s)he is a white collar worker? ¢ ¤ RW DW IW RB DB IB    § ¨ ©£ § ¦ ¦  ©£ ¨ ¥ ¦¨  ¢   ¢  "!  § ¥ ¦¨    § § ¤£ ¦ § § ¦ § ¤£ ¦   ¤ ¥£  £ § ¥  ¢  ¦ ¥ ¦¨  ¢ ¢ ¢ § § ¦ £ ¤£ ¦  ¦  ¤ ¥£ £ ¤£ ¢ ¢ ¢ probability that (s)he is a blue collar worker?   If a person is NOT a Democrat, what is the ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 74 EXAMPLE 3.13: (p. 123) develops cancer § individual smokes  § ¤ Given: Simple Events Probabilities  !   "¥    ¤  !     ¤  ¤ ¥ §  "¥     ¦  §   ¤ ¥£ ¢   ¦  ¦ ¤ ¥£ ¢ © ¨ ¤  ¤ ¥£ ¢ § ¦ ¤ ¥£ ¢ §  ¥ §   !   ¦ § ¤ ¥£ ¢ ¦ given that the person smokes. ¥  Find the probability that a person develops cancer  ¦  ¤ ¥£ ¤ ¥£ ¢ ¢ § ¦ ¤  ¦  "¥    £ ¤ ¥£ ¢ § ¢  ! ¥ § § ¦ ¤  £ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 75 Multiplicative Law of Prob. (p. 128) ¦ ¦ ¤  £  ¢¦ ¤ ¥£ ¢¦ ¤ ¥£ ¢  £ § ¢ ¦   ¤ ¥£ ¢ § Why? Recall (Defn of Cond‘l Prob.): ¦   ¦ ¤ ¥£ ¤ ¥£ ¢ § ¢ ¦ ¤  £ ¢ so ¦   ¤ ¥£ ¢ § ¦  ¦  ¤ ¥£ ¤ ¥£ ¢ ¢ ¦ ¤ ¥£ ¢ § ¦ ¤  £ ¢¦ ¤ ¥£ ¢ Useful for computing the probability of the intersection when : ¡ Conditional Probability is given ¡ Conditional Probability easy to get without definition. STA 2023 c B.Presnell & D.Wackerly - Lecture 6 76 EXAMPLE : (#3.61, p. 137) Microchip:   ¨ § ¦ fail on first use £ ¢ If it lasts through its first use, prob. that it lasts a year ¡  is . Find the probability that a microchip does not fail during the first year. NOTE: If chip does not fail during first year–It cannot fail on its first use!!! Microchip lasts through first use , ¡ .   § ¡  ¢ . ¨  ¤ § ¦ ¤ ¥ §  ¤ ¥£ § ¦ ¤ ¡ ¤ ¥£ ¤ ¢ ¡ §  £  ¢ ¡ “Does not fail during first year”   § ¦ Chip lasts from first use through first year ,  ¢ £ ¡ £¦  ¤ §  £ ¢ § ¦   ¤ ¥£ ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 77 EXAMPLE : (Screening for quality) Shipment of 30 cameras, 6 defective. Choose 2 at random. If both good, accept shipment. Find the prob. accept shipment. ¤ ¥   § ¥ § ¦ ¤ ¥£ : £ ¢ ¡ ¢ camera good ¨ § ¤ If first camera known to be good, what is (cond.) prob. second is good? § ¦ ¤  £ : ¨ ¢ camera good ¦¤ §¥¥ §  Start with 30 cameras, 6 bad. If first chosen is good, when we select the second, ¡ there are 29 cameras, 23 good, 6 bad.  ©  .   !  ¥ ¥ shipment accepted ¡ £ ¥  § ¦ ¤  £ ¢¦ §  © § ¤ ¥£ ¢  ¦ §  ¤  £ ¤ ¦ ¢ ¡ Note that   ¤ ¥£ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 78 Independence Does knowing that one event has occured alway have an impact on the probability of another event? ? ¦ ¤ ¥£ ¢ § ¦  ¤ ¥£ ¢ ¡ Is EXAMPLE : Toss a balanced die. ¢ § ¡  ¦   ¡ § ¢ ¥ ¡ ¡ £ ¡ §  ¡   ¥ ¨ ¡ § # greater than 2  # greater than 3 ¡ § get an even number £ and ¢ ¡ ¡ § ¤ ¡ §  ¡ ¤ § ¢ £ ¥ § ¦ £ © ¢ and ¤ § £ ¦  § £   ¦ ¤ ¥£  £ ¢ ¢ , © ¡ ¦  §  ¤ © £ § ¦  ¤ ¥£ ¢ ¦ ¡ § £  ¦ § ¤ ¥£  ¡ ¢ ¡  ¤ ¤ §  ¤ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 79 ¥  © §  ¦ £ §  £   ¤ ¥£ ¦  £ ¢ § ¢ ¦  ! ¤ ¥£ ¢ ¤  ¨ © § ¥ § £  ¤ ¤ ¥£ ¢ ¤ § ¤ ¦ ¥ and  ¦ £ £ ¢ ¦ ¤ ¥£ ¢ DEFINITION: (Def. 3.9, p. 131) Events are said to be independent if  ¦  £ ¢ § ¦ ¤  £ or ¢ ¦ ¤ ¥£ ¢ § ¦  ¤ ¥£ ¢ “Not independent” = “DEPENDENT.” Ex. Die Toss: are independent then ¦ ¤ ¦  £  £ ¢¦ ¢¦ and  ¤ ¥£ ¤ ¥£ § ¢ ¢ § ¦ ¤ ¤ Note that if INDEPENDENT. ¤ ¤ and DEPENDENT,  and [by Mult. Law] [by independence]   ¤ ¥£ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 6 80 This is another way to define independence: (p. 133) Independent  ¦  £ ¢¦ ¤ ¥£ ¢ § ¦   ¤ ¥£ ¢ ¦ WARNING Do NOT confuse INDEPENDENT with MUTUALLY EXCLUSIVE. sets have no points in ¡ ¡ Mutually Exclusive common. ¡ Independent – need to check PROBABILITIES. ¡ Suppose that and that  § ¦  £ ¢ ¨ § ¦ ¤ ¥£ ¢ §  ¦  £ ¢ § ¦   ¦ ¤ ¥£  £ ¢ § ¢ ¦  ¤ ¥£ ¢  ¤ ¥£ ¢ § ¨ §  § are DEPENDENT!! ¦ ¦   ¤ ¥£ and  are mutually exclusive ¤ ¢ ¤ ¢ ¢  and , STA 2023 c B.Presnell & D.Wackerly - Lecture 6 81 Ex. (LAST TIME!!!) Odds are 2:1 that when H and T play racquetball, H will win. H and T play 3 matches, and winner of each recorded. Odds are 2:1 that H wins means H wins with prob.  £ ¨ . T wins with prob. . Probs. given  £ ¥ § ¦¨  ¥£ £ ¥ in the example for 8 possible outcomes were calculated assuming matches independent : £ ¥  ¥  £ ¥ ¥ ¥  £ ¥ ¥ £ ¥ ¨ ¥ ¨ £ £ § ¦ £ £ £ £ ¨£ £ £ £ ¦ £ £ ¨£ £ £ £ ¦ £ ¦ ¨£ £ ¨£ ¦ ¨£ £ ¥£ ¦ £ ¥£ £ ¦ ¥£ £ £ ¥£ ¦ ¥£ ¦ £ £ ¨£ £ £ ¥£ ¦ ¨£ ¦ ¥£ £ £ ¦ ¨£ £ ¦ ¥£ ¦ ¥£ £ £ ¦ ¨£ ¦ § £ £ ¥£ ¦ ¥£ ¦ § ¥ £ £ ¦ § £ ¦ ¨£ ¦ § £ £ ¦ § £ ¥£ ¦ § £ ¦ £ ¨£ ¨£ TTT § TTH ¤ THT ¥ THH £ HTT £ HTH £ HHT £ HHH Probability Outcome STA 2023 c B.Presnell & D.Wackerly - Lecture 6 and are  ¤ EXAMPLE : Like #3.52(a), p. 136 82 independent events: §  . ¦  ¦  £  ¨ ¢ ¤ ¥£ ¢ ¦  £ (independence) ¦ ¦ £ ¢ ¢ ©¦ ¦ §  £ § ¦ ¦ ¤ ¥£ ¦ ¥  §   § ¨  © ¨   § ¤ ¥£ ¢ ¦   ¢ ¤ ¥£ ¢ § ¨ § ¥ ¢ ¤ ¥£ ¢ ¤ ¥£  ¢  ¢ £  ¢ ¤ ¥£ ¨  ¢ ¤ ¥£ £  ¤ ¥£  ¢ ¨  ¤ ¥£ ¦ § § ¦ ¨ ¥  ¦ (Add. Law)   ! ¦  £ (independence) ¦ § ¥ and ¨ Find §  ¤ ¥£  ¢ ¤ ¥£ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 83 Thought: Whether you think you can or you think you can’t, you’re right! Assignments : Today : p. 163–169 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, 179-183 For Tuesday:Exercises 4.22, 4.27-29, QUIZ 2 – Chapter 3 – Project 1 Last Time : ¡ Multiplicative Rule (p. 128): ¤ ¥£ ¢ £ ¦ ¤  £ ¢ § ¦  £ £ ¢ ¦  ¤ ¥£ ¢ § ¦  Independent (p. 131, 133):  ¦   ¤ ¥£ ¢ and ¤ ¦ ¤ ¥£ ¢ § ¦  ¤ ¥£ ¢ ¦  £ ¢ § ¦ ¤  £ ¢  ¦  £ ¢ £ ¦ ¤ ¥£ ¢ § ¦   ¤ ¥£ ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 84 EXAMPLE : Space shuttle Challenger disaster. The Rogers commission concluded disaster results unless all six field joints (sealed by O-rings) work. ¡ first field joint works ¢ § . . . ¡ sixth field joint works £ § Assuming joint failures indep.: ¢  ¥ £ £ ¢¦    © £ ¢ § ¦ £ ¢ ¢ £ ¢¦ £ ¥ ¡ F), estimate ¦  ¨ ¢¦ £  ¢¦ £ © £ ¢¦ ¢ § £  § ¦ , so £ ¤ ¥ § ¦ £ £ ¢ § ¢ ¢¢ ££¢ § § ¦ ¦ £ £ ¢ ¢ ¤ ¢ ¦ £  , only ¦ § § ¦ ¨   § £ £ ¢ ¦ § ¤ ¢ F. ¡ ¨ ¡ ¨ £ ¢ ¢ ¢¢ ££¢ ¡ £ © § ¨ ¦ § £ ¦ ¢ ¡ ¦   -th of what it was at © § ¢ ¤ £ ¢ about ¤ at least 1 fails § ,  ¨ Note: At at least 1 fails  all six work £ Under Challenger flight conditions ( ¦ all six work STA 2023 c B.Presnell & D.Wackerly - Lecture 7 85 EXAMPLE : (Like # 3.109, p. 158) : HIV test. : person tests positive : person tests negative : person HAS HIV : person does not have HIV  § §  Suppose we know that “sensitivity of test” ¤ ¢ “specificity of test”  § § ¦ £ ¢  ¦ §  § £ ¢ If (1996 CDC) 476,999/242,200,000 = .00197 is the fraction of pop. that has HIV, then for person selected at random  ¤  ¢  ¢ § ¦ § ¤£ ¢ ¦ ¨    § § § ¦ § ¤£ ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 86 A person is selected and tests positive. What is the (cond.) prob. that the person actually has HIV ? £ ¢ , and  § ¦ £ ¢ , § ¤£ ¦ ¢ § ¤£ ¦ ¢ ¡ ¡ § ¤£  ¢ § ¦ £ ¢ § ¤£ ¦ ¤ ¡ ¢  § § ¦ £ ¢ ¡  § § ¦   £ ¢ ¡ Since ¦ Know : (WHY?) § Want , ¦ Know . ¨ !  § ¦ § £ ¢ ¡ § ¤£ ¦ ¦ ¢ £ ¨  £  § ¦ £ £ ¦  ¤ ¢ ¨  !  § § ¦ £ §  § ¤£ ¢ . § ¤£ ¦ § ¤£ ¢ , found ¦ § ¨ ¥  ¨  ¦ £ ¢ ¨ ¨   ¨   £ § ¤£  ¡ § ¢ § ¦ ¢ § ¤£ ¦ ¢ ? ¦ KNEW § ¤£ ¦ ¢ ¡ How about ¨ ¤   ¤ £ ¢ ¢   ¢ £  £   ¨   § ¦ § ¤£  ¢   ¦ ¨ ¨   § ¨     § ¨    ¢ § ¨   ¦ § ¤£  § ¢ § ¦ £ ¢ Thus §   and are mutually exclusive § ¤£   ¦ §  ¦ § ¤£  § Note that ¦ £ ¢ ¡ How about ? STA 2023 c B.Presnell & D.Wackerly - Lecture 7 87 STA 2023 c B.Presnell & D.Wackerly - Lecture 7 88 Chapter 4 : Discrete Random Variables Intuitively : a random variable is a quantity that ¡ takes on different values (the “variable” part) ¡ 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”, © or ¨ ¨  Payoff . ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 89 ¡ CONTINUOUS (Def. 4.3, p. 166) – the variable can take on all values on a line interval (look for variables where the possible values change smoothly, without jumps) – Time to assemble automobile – Height, Weight , Blood Pressure – Amount of rainfall (in inches) in a month Def. A RANDOM VARIABLE (Def. 4.1, p. 164) is a rule that assigns one and only one numerical value to each sample point. Ex. Roll two dice. 36 sample points. Let sum of up faces. ¦¨ ¦¥ ¦ ¨£ ¨£ . . . Sample Point 2 3 . . . 12 £ § STA 2023 c B.Presnell & D.Wackerly - Lecture 7 90 DEFINITION : (Def. 4.4, p. 169) The probability distribution of a 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 (grey shaded box, p. 169): for all poss. values ; (sum is over all values of ). ¨ £ ¡ ¨ § ¦ ¡ ¦ ¡ £  ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 7 91 Ex. Shipment of 30 cameras, 6 defective. Select 2 at random. # defectives selected (0, 1, or 2). § : 1st camera selected is defective : 2nd camera selected is defective ¤  ¦ ¤   £ ¢¦ ¤ ¥£ ¢ ¦ §    " ¡¦   ¤ ¥£ ¦ ¦ ¦  ¨   §  ¦ ¤ ¥  £ £ ¢¦ §     ¥£¦   ¤ ¥£ £ § £ ¥ £ § £ £  "¥ £ ¦   § ¦ £ ¢  ¥£ § ¢ § ¢ ¦ ¦¨ ¦  ¦ ¢ ¦ ¢ ¦ ¤ ¥£  ¤ ¥£ ¤ ¥£ ¤  ¥  £ £ §  £ ¢¦  £ ¦   § ¤ ¥£ ¢ £ ¤ ¥£  ¥£ ¢ § ¦¨£ ¢ § § STA 2023 c B.Presnell & D.Wackerly - Lecture 7 92 § ¦   ¤ ¥£  " § ¢ ¦¥ § ¦ ¥ £ § £ ¦   ¦ £ .0345 ¢ 2 ¤ ¥£ .3310 ¢¦ 1  £ .6345 ¤ 0 ¦ £ £ £ § ¦ ¥£ ¢ § Table of the dist. of : STA 2023 c B.Presnell & D.Wackerly - Lecture 7 93 Ex. Sum of up faces on two dice. (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) Sample Points (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) £ (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) ¦ (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) 2 (1,1) 1/36 3 (1,2), (2,1) 2/36 4 (1,3), (2,2), (3,1) 3/36 5 ... 4/36 6 ... 5/36 7 ... 6/36 8 ... 5/36 9 ... 4/36 10 ... 3/36 11 (5,6), (6,5) 2/36 12 (6,6) 1/36 STA 2023 c B.Presnell & D.Wackerly - Lecture 7 94 Distribution of the Sum of Two Fair Dice p(x) 6/36 5/36 4/36 3/36 2/36 1/36 x 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
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