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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: MultipleChoice 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: TrueFalse 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 “373 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 qQtecLloquY 5]: HO.E_D¢+3QM 43°, iplqy‘]
V ' :L—IO, l8$1+ZO'51
3%
:; Ho ’62 _ 63 138 _i—$2.lll Chapter 5 — Expected Value 5.2 Using FiveStep Simulation to Estimate Mean Values Example 9: LeftHanded vs RightHanded
Approximately 9 out of 10 Americans are righthanded and 1 in 10 is lefthanded. If we
randomly select 30 Americans, how many will be left—handed? Use the fivestep 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 ‘ ‘.
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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.
 Spring '09
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