2-23 - CHAPTER 5 — EXPECTED VALUE AND SIMULATION 5.1 The...

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Unformatted text preview: CHAPTER 5 — EXPECTED VALUE AND SIMULATION 5.1 The Expected Value (Theoretical Mean) of a Random Variable The theoretical mean of X (or the expected value of X) is the average value of X in a very long series of independent repetitions of the random phenomenon. Example 1: Multiple-Choice Test Claire is taking a multiple—choice test that has 50 questions. Each qUestion lists five possible answers, but only one is correct. Claire has not learned the material and does not have time to study before the test. How many correct answers should she expect to get? I E [number d’p Correaf Of) 6003(51 t: #o‘fl?ug§‘é{ofl S - P(r‘$%h—bqnswer> :5O'EZE Law of Large Numbers for Independent Repetitions In a long sequence of independent repetitions of the random experiment, the sample mean )7 of the observed values of X will be very close to the theoretical mean ux. Or v Y z #X. This is also true if we have a large sample size and we’re sampling at random without replacement from a real—world population with mean ,uX. Example 2: Spades — Five cards are dealt one at a time from a well—shuffled deck of 52 cards. Sam wants to know, on average, how many spades he will get per hand. In other words, find E (X). “I. 010 farce: 4 PC3P¢QJ€> ; 5 . T",— :lLZf—il In both of the previous examples, the variable of interest is a counting variable. Expected Value of a Counting Random Variable Consider an event that has probability p of occurring in each of n trials, the expected value E (X) (theoretical mean fix) is given by E(X) = ,uX= np. Chapter 5 — Expected Value Example 3: True-False Test Edward is taking a true—false test that has 30 questions. Edward has not learned the material and does not have time to study before the test. How many correct answers should he expect to get? EEJ‘WP. :IE' If the random variable has muitiple possible values with corresponding probabilities, the expected value of X can be computed as follows. Expected Value of a Random Variable If the possible values of X are x1, x2, ..., xk, viiith corresponding probabilities p(x1), ..., p(xk) the expected value can be computed as ' E(X) = Ux = X1P(X1) + X2P(X2) + + XkP(Xk) = Z XP(X)- Example 4: Rolling a Die What is the expected value of a roll of a die? Home will 1—; + 2.: + 3'2: + Li=t+lsé+<oZ :é(liz+3+H+5+é> 1 itzfiaiacfl Example 5: Spinner What is the expected value of a spin for the spinner shown below? , ___I__ 00 ,i Co (a H 5,22.-. ' a*§*a*a*a’ 8 “37-3 62 Chapter 5 — Expected Value Expected Value of the Sum Y of n Random Draws from a Box If Yis the sum of n repeated draws from a box (with or without replacement), then E(Y) = W = n - (mean of box) Expected Value for the Sample Mean of n'Random Draws from a Box The expected value of the sample mean )7 of n repeated draws from a box (with or without replacement) is ' E(X) = ,u— = box mean. X Example 6: Board Game For a particular board game, players roll a die and'then advance the number of spaces shown on the die. After 50 rolls, how many spaces would you expect to advance? E [5MM G‘P ED Foll‘é ofladccl '5 5O ‘ E EC“: (0” z 50. 3.5 = How many spaces would you expect to advance after 100 rolls? ‘ _ E [sumac zoo rolls 07C Ace} : ma fiche rail] ' ‘ "r 100 3 . 5 2 f ’5 150 | Example 7: Survey A survey question asked 400 Stat 100 students “On a scale of 1 —— 10, 10 being the strongest, how strongly do you believe in ghosts?” If the average rating was a 6, what average rating would you expect for a sample of 300 students? [EL—face} : ML!» ‘” Example 8: Roulette Wheel A roulette table has 38 slots — 18 red, 18 black, and 2 green. If you bet $1 on red so if it comes up red, you win $1. Otherwise, you lose $1. If you play 40 times in a row, what is the expected value of your winnings? ' [/8 iii 20 Etne'lrcaqm qQ-tecLloquY 5]: HO.E_D¢+3QM 43°, iplqy‘] V ' :L—IO, l8-$1+ZO-'51 3% :; Ho ’62 _ 63 1-38 _i—-$2.lll Chapter 5 — Expected Value 5.2 Using Five-Step Simulation to Estimate Mean Values Example 9: Left-Handed vs Right-Handed Approximately 9 out of 10 Americans are right-handed and 1 in 10 is left-handed. If we randomly select 30 Americans, how many will be left—handed? Use the five-step method to solve this problem. Step 1. Choose a Model - We can either use a box model or a table of random digits. Let’s use the first method. (See page 255 for an example of how to solve this problem using the second method.) Illustrate an appropriate box model for this scenario. What is one simulation for the box model you set up in step 1? {BED w/ replacement Step 3. Define the Statistics of Interest What statistic are We interested in investigating? What in our box model corresponds to this statistic of interest? « WC; tells ULS “Kiln? # (Mg /e{£;lflanoieo/ / ‘ r m [a Step 4. Repeat the Simulation (Pea/D 6 ' '4 <9” ‘3" P Using the computer software, set up the box model and run the simulation 10,000 times. ' Step 5. Find the Mean of the Statistic of Interest Based on the results from the simulation, what is the expected number of left—handed people in our randomly selected sample of 30 Americans? 2.47/2 %3 64’ ‘ v . ‘ < ‘ ‘l ‘ ‘ ’ 7 ' K . . .l< ' . J 3‘- ‘ . . r '1 ‘ ‘. . . i . _ .. ,.- . .. . r ,A . .1 r ., ,,.-:. ., .V . 2., _, w , 'w s v i .. a, ', a, m: ~ A 2K K 21 Q L‘ s t)’ v: f A; «9 s .:v a: f? «i :’ 3. :3 s, l J. 6 '25 : nuns ; qr" \ J, 0% v. ‘ . . .( ~ :3 . A; ._ ... l .3 . (I , u, \, ,, , I ‘ V, , a a ham» w W2 M2 s “é fi :. 3 w ; 2‘“ / L » 3 1; , w. e. A .; , . . g r _ , 4_ _» at. , r d, . s, . ,, .. . _ . . .. I .. . , . ‘v I ‘ a ...
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This note was uploaded on 08/08/2009 for the course STAT 11 taught by Professor Hirtz during the Spring '09 term at University of Illinois at Urbana–Champaign.

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2-23 - CHAPTER 5 — EXPECTED VALUE AND SIMULATION 5.1 The...

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