Week5 - STA 2023 c B.Presnell & D.Wackerly - Lecture 8...

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Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 8 Thought: There are three kinds of people: those who can count; and those who can’t. Assignments : For Today: pages 172–176 For Tuesday: Exercises 4.22, 4.27-29 QUIZ #2 Covers Chapter 3 and Project 1 For Wednesday: Read pages 179–185 For Thursday: Exercises 4.33, 4.36 95 STA 2023 c B.Presnell & D.Wackerly - Lecture 8 96 Last Time : ¦ ¢ ¡ ¨ © . What is £ ¥ ¦ §¥ £ ¤¢ ¡ ¨ and ¦ §¥ £ ¤¢ ¡ ¨ (F. S.) ¨ ¦ ¢ ¡ ¦   ¨  ¨ (F. S.) ¨© ¦ £ ¤¢ £ ¤¢ ¦ §¥  ¡  ¡ £ ¤¢ ¡ £ ¤¢  $ ¨ ¨ ¨ £ ¥ ¦ ¦ ¦ ¢   £ ¤¢ ¡ £ ¤¢    HIV Example (Like #3.109, p. 158) ¡  £ " #!  mutually exclusive ¦ ¨© ¦ ¢ ¨© (F. S.)  ¡ £ ¤¢  ¡ ¨© %¨ ¦ &¥ ¦ £ ¤¢ ¡   £ ¤¢ ¡ ¨© ¦   ¨ £ ¤¢ £ ¤¢ ¡   ¡  Put the pieces together! Discrete and Continuous Random Variables (p. 166). Probability Distribution for Discrete Ran. Var. (p. 169) ¨ Know ? STA 2023 c B.Presnell & D.Wackerly - Lecture 8 97 EXAMPLE : Have 3 light bulbs, 2 good, 1 defective.  ¢¡ ¨¢ ¡ means select first, then ¨ ”Label” the bulbs # defective. ¨£ ¡£ ¥§¦ ¥¤¢ ¡ Randomly select 2, second. ¢ ¦ ¡ ¡ ¢¡¨ ¨ ¦ ¡ ¦¡¨ ¦ ¢ ¡ ¡  ¨ ¢ ¡ ¢ ¨  ¢   ©  ¨  ¢  ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 8 98 RECALL: (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: ¢ ¨ ¡ for all poss. values ; ¢ .  ¢ ¨ ¢   ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 8 99 Ex. Raquetball: odds are 2:1 that when H and T play, H  will win. H and T play 3 matches. # matches won by H. Sample ¢ ¢ ¨ ¢ ¢    ¨ ¢ ¨  ¨ ¢ ¨  ¨ ¢ ¢ £¤¢ ¡ £¤¢ ¡ £¡ ¤¢   ¨ ¢ ¢ ¢  TTH ¢ ¢ ¢ THT ¢ ¨ ¨ £¥© ¡ £¥¢ ¡ £¡ ¥¢  THH  ¢ HTT ¨ £¤¢ ¡ £¤© ¡ £¡ ¤¢  HTH ¨ ¡ © ¨ ¡ ¢ ¨ ¡ ¢ ¨ ¢§¡  ¨ ¡ ¢ ¨¢§¡  ¨¢§¡  ¨ ¢§¡ ¦ HHT £ Probability HHH ¨ £¡ ¤© ¢ ¢  ¨ £¡ ¥ ¢ ¢  ¨ £¡ ¤© ¢ ¢ TTT Point STA 2023 c B.Presnell & D.Wackerly - Lecture 8 100 The Expected Value a Random Variable Consider the lightbulb example. Repeat experiment a large number of times, say 600. Expect times;   ¨  ¨ ¡ ¢¢ ¦ ¡ ¢¢   ¢ So, average value of roughly ¢ roughly times over 600 repetitions should be about £ £ ¢ ! % ¦§¦§¦ $ ££ ¥¤¦ © % ¢ " %   £  ¨ ¢ ¢   ¨ % ¢ %   ¡ ¢  £ ¡ % ¦¨§¦ ¦ #$ ¡ ! ¢ ¢ % ¢ % ¨  ¢ £   ¨  ¢  " STA 2023 c B.Presnell & D.Wackerly - Lecture 8 101 Def. (Def. 4.5, p. 172) For a discrete r.v. , the expected value of (population mean of ) is defined to be ¨ ¢   ¨ ¢   ¨ is a “weighted average” of the possible values ¨ is NOT necessarily a possible value of .  ¢ of . ¢  STA 2023 c B.Presnell & D.Wackerly - Lecture 8 EXAMPLE : Suppose # of calls to an office in a 5 min. ¨ period has distribution 102 ¢  ¢  £ ¢   ¢  £ ¨ ¢      STA 2023 c B.Presnell & D.Wackerly - Lecture 8 103 EXAMPLE : #4.27 p. 177 Rock concert: No rain, profit of $20,000, Rain, lose $12,000 profit. ¨ ¢  .     ¨ rain ¨    ¢  ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 8 104 Def. (Def. 4.6, p. 174) The variance of a discrete r.v., , is ¢  ¦ ¨ ¢ ¢ ¦  £ ¤¦  ¨ © ¢ ¨  ¢ ¡ ¢ ¢ ¢ ¢ ¦  § ¨¦ (Def. 4.7, p. 174) is ¥ . is ¦ ¢ ¨ ¢ ¦  A shortcut formula for the variance of ¦  The standard deviation of ¨ Recall that  ¦  ¦ ¢ ¨ ¦ ¢   ¦ STA 2023 c B.Presnell & D.Wackerly - Lecture 8 105 ¨ EXAMPLE : Telephone Calls ¢  ¢  £ ¢   ¢  £ ¡ £¦ ¨ ¢   ¢ ¡ ¢ ¦       STA 2023 c B.Presnell & D.Wackerly - Lecture 8 is  and ¡ Interpretation of 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 – The “at least” – Tchebysheff. – The “approximately” – Empirical Rule 106 STA 2023 c B.Presnell & D.Wackerly - Lecture 8  ¢  £  £ ¦ ¡    ££ £ £ ¢  £  ££ ¡   £ ¨ ¨ £    ¦ ¨  ¢ ¢ £ £   ¢    ¡ ¢ ¢ ¡ ¡ ¢ or  ¦ ¢ and  ¨  between   ¡  ¨ ¦  or  ¡ and .  £  between are Possible values of ¦ EXAMPLE : Telephone calls 107 ¡  ¢  ¢ ¡  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 costing $10 will pay $1 million if the passenger dies in a crash. Let be the “profit” to the insurance company from a single policy. Estimate prob. of fatal crash to be . ¡ ¨  ¢  £ £ ¡ ¢¦ ¨   ¢  STA 2023 c B.Presnell & D.Wackerly - Lecture 9 109 Thought: We all have photographic memories–some of us just don’t have film. Assignments : Today : Pages 179 – 185 For Thursday : Exercises 4.33, 4.36 For Monday : OPTIONAL REVIEW For Tuesday : EXAM 1 STA 2023 c B.Presnell & D.Wackerly - Lecture 9 110 Last Time : Mean p. 172 ¨ ¢   ¨  ¢  Variance p. 174   ¦ ¨ ¢ ¢ £¦  ¨ ¢ ¡ ¢   ¦ EXAMPLE : Salesman contacts 1 or 2 customers per ¦ ¢ and day. Probabilities are , resp. Each contact results in £ ¢  Find the distribution of £ $50,000 sale – probability is ¢ ¢ no sale – probability is daily sales. ¦ ¢ (p. 174) is ¨ The standard deviation of . STA 2023 c B.Presnell & D.Wackerly - Lecture 9 111 A sample point must indicate: How many contacts were made. Sale(s) made. ©  1 contact, sale  2 contacts, sale on 1st, no on 2nd Sales £ £ Probability ¢ ¢  ¢  ¢ £ ¢ © Sample Point © ¢ £ £ £ ¡ ¢¢  £ ¥£ £ ¥£ ¦¢ ¢  £ ¥£ £ © © ¢ ¡ ¢¢ ¢ ¢ ¢ £  £ £  ¢ £ ¦ ¢ £ ¢ © ¦  ¢  ¢ ¦ © £ ¨ ¡  ¡¡¡ £ ¡¡¡ £  ¢ ¢ ¢ ¡  ¦ ¦¦££ £ ¡¡ % £ £ ¦£  ¢ ¡  £ ££££ ££££ £¦£ % ¡   ¦ ¢ ¨  ¢ ¦  ¦    ¨ ¨ ¢   ¢  STA 2023 c B.Presnell & D.Wackerly - Lecture 9 112 STA 2023 c B.Presnell & D.Wackerly - Lecture 9 113 Thus, for the sales example, £¦ ¨ £  What is the probability that    ¢  is in this interval ? STA 2023 c B.Presnell & D.Wackerly - Lecture 9 114 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. ¨ ¡ ¢ ¡  ¢  ¢ ¨ ¢ © ¢ ¡ © 3. Prob. of © or   © 5. Variable of interest is , the number of ’s in the trials.  ¢   , . ¡  ©  £ ¡ ¢  # H in 10 tosses, , ¤ Ex. Toss a fair coin 10 times, .  Ex. Choose 10 microwave ovens at random from © factory output. Count number defective ( =defective,  ¡ =good). unknown. 4. Trials are independent. STA 2023 c B.Presnell & D.Wackerly - Lecture 9 115 © Ex. Play three games of racquetball =Henry wins, ¡ =Thomas wins Ex. Diagnostic device : detects or misses Ex. Political Poll : Likes Bush, Not Objective : Obtain a formula for the probabilities associated with all values of a Binomial Random Variable, . –The Probability distribution of . Some Preliminary Results is a positive integer, , read ¢ factorial is Def. : (p. 139) If  ¨§¦ ¦¦  ¨ ¢ ¢  ¨ ¢ ¢ ¢   ¢      ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 9 116 objects , where the order in which the from among Fact: The number of ways of selecting objects are selected does not matter is   ¡  if ¨ ¢ ¢  choose .  is read  ¡ if .  EX. : Club consists of 10 members, select a committee of .    size £ size ¢      ¢ ¢ : ¢  STA 2023 c B.Presnell & D.Wackerly - Lecture 9 trials and . £  £ £ ¢  ¡§ for  £ ¢ are £ £ ¦ ¢ ¢ ¡   Possible values of £ © ¨ . (p. 183) is a binomial random variable with If 117 .   ¨ ¢  Why this formula? Sample point approach: £ – toss 3 coins, look at 3 microwaves, etc. Sample Points Probabilities ©©© ©© ¡ ¢ ¦ ©     ¦ ¦ ¢ ¦  ¢ ¡ ¡ ¡  ¢ ©© ¢ ¡ ¢ ¢  © ¡  ¢ ¢   © © ¡ ¡ ¡ © ¡ ¡ ¡ Ex. : STA 2023 c B.Presnell & D.Wackerly - Lecture 9 118 Note distinct sample points . probability  Each single sample point where  corresponding to There are has .  ¢  ¨ Thus Try the formula .  £     ¨ ¢  ¨  ¢  £    Thus,  Same as sample point approach!! STA 2023 c B.Presnell & D.Wackerly - Lecture 9 119 For any sample point of a binomial experiment, in ¢ ¡ for each . successes each have ¡ ¢ ’s and  the sample point, and a Sample points that give for each © prob. is calculated by multiplying a ’s in some order, © so each sample point corresponding to the value has prob. . How many such sample points are there? $ © ¡ ¦ positions of the © positions to be filled © ¡ with an "! ¦¦ ¦¦ §§¦ §§¦ – Think of selecting ¡ – One sample point : , and then filling in the rest as – There are ’s. ways to do this, so there are sample points corresponding to the  ¨ ¢  Thus  value . STA 2023 c B.Presnell & D.Wackerly - Lecture 9 120 Ex. H and T play 3 matches of racquetball. H has prob. 2/3 of winning any match. Matches indep. Probability  ¨  ¨  ¨¢ ¡ ¨¢ ¡ ¨¢ ¡ ¨ ¡  ¨¢ ¡ ¨ ¡  £¡ ¥¢ ¨ £¡ ¥ ¢ ¨ ¢ £¡ ¥¢  £¡ ¥¢ ¢   ¨ ¢ ¢ £¡ ¥ ¢ ¢ £¡ ¤¢ ¨ ¢ £¡ ¤© ¨  ¢ ¨ ¨ £¡ ¤¢  £¡ ¥ ¢  ¢ ¢ £¡ ¥ ,  # won by H.   £  £ sample points with 3 2 1 2 1 1 0 There are   , ¨ ¡ ¦ HHH HHT HTH HTT THH THT TTH TTT Outcome . 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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