LM /23$ ArialY.@:g#3#@$ArialY.@:g#3#@$ ArialY.@:g#3#@@TMonth' 'D', 'YYYY Normal Comma Currency Percent Fixed DateHeading 1Heading 2 Total Comma0 Currency0 033 R033 2033 B033 033 x033 033 033 0 33 P033 0033 033 033 033 @33 P33 `33 @33 P33 `33 33 33 `33djY!_]<  ?00 FDQ$  A   ?a      H`?bQ4V@EX,b PPH\ 6HeRd]/L!R|_G)v-Hdp),S6RiT*ʦȤ} t*;%gZmVe>"{#p0|[Mb@ ]c6Go9<agtZ=![{u?4^utm !ebnR"-F[aeS&/ qm[XPqЀ<n=W ˃tL{0/Yjk+{r9/ ~ȾI|?@>"{|?0&kc@Q Ebg Qtv&BȇDjh ?30 z ' R'ʒ)̱-R~LS|'~_3 gɠ}9̓y8#3 A=H<͔Eh4֋OA4I Jr=O EQԕ-MSMUUՕm]WeY֕l *' """ "  he W Wind ecte        fv$        Mq  CU< X1Axis? Ms  {C% Y1Axis?  Mt  d2X,  Y2Axis? vj  CU 1PaneLeftWallRightWallBaseSeriesSeriesLabel?  Rm  d2X,  X1Title? C@Ro  d2X,  Y1Title? l@Rp  d2X,  Y2Title? @Ml  d2X, Legend? }LZ  $ Title?$ArialArial "]Z   =%&'( @@0d  +hR$ Arial:g##@:g#$ Arial:g##@:g#  LB  %&'( @@0d  $ Arial:g##@:g#$ Arial:g##@:g#  B    ^ ddư> OR?& Mb@?  #  f g i2  -./g 0d |  0B00 L6$u<7~7G[\B       @'SM230 - Spring 1997, Sums of Random Variables, J. Turner ` `'Centery'SUM(P(X))'SUM(P(Y))SӇ'SUM(P(W))6 `$'See sheet B for instructions"'See sheet C for assignmentB'To get started, Press Home, then End, then the down arrow.B'If you get lost, press these 3 keys again to get centered. ` ` ` `  `  `  `  `  ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `  ` !` "` #` $` %` &` '` (` )` *` +` ,` -` .` /` 0` 1` 2` 3` 4` 5` 6` 7` 8` 9` :` ;` <` =` >` ?` @` A` B` C` D` E` F` G` H` I` J` K` L` M` N` O` P` Q` R` S` T` U` V` W` X` Y` Z` [` \` ]` ^` _` `` a` b` c` d` e` f` g` h` i` j` k` l` m` n` o` p` q` r` s` t` u` v` w` x` y` z` {` |` }` ~` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `"P "P5E"P?9 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `  `  `  `  `  ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `  ` !` "` #` $` %` &` '` (` )` *` +` ,` -` .` /` 0` 1` 2` 3` 4` 5` 6` 7` 8` 9` :` ;` <` =` >` ?` @` A` B` C` D` E` F` G` H` I` J` K` L` M` N` O` P` Q` R` S` T` U` V` W` X` Y` Z` [` \` ]` ^` _` `` a` b` c` d` e` f` g` h` i` j` k` l` m` n` o` p` q` r` s` t` u` v` w` x` y` z` {` |` }` ~` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` h q"X &@   $@   "@    @   @   @   @   @   @   @   ?                            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" # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~                                                                                                                                                      h                             !  "  #  $  %  &  '  (  )  *  +  ,  -  .  /  0  1  2  3  4  5  6  7  8  9  :  ;  <  =  >  ?  @  A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z  [  \  ]  ^  _  `  a  b  c  d  e  f  g  h  i  j  k  l  m  n  o  p  q  r  s  t  u  v  w  x  y  z  {  |  }  ~                                                                                                                                                                h                             !  "  #  $  %  &  '  (  )  *  +  ,  -  .  /  0  1  2  3  4  5  6  7  8  9  :  ;  <  =  >  ?  @  A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z  [  \  ]  ^  _  `  a  b  c  d  e  f  g  h  i  j  k  l  m  n  o  p  q  r  s  t  u  v  w  x  y  z  {  |  }  ~                                                                                                                                                              $u<7~7[\  ܥ?=3@'SM230 - Spring 1997, Sums of Random Variables, J. Turner'InstructionsH'1. In the red cells, enter the probabilities associated with X.H'2. In the blue cells, enter the probabilities associated with Y.<'3. The probabilities for X+Y are in the green cells.d'4. Under the cell in the first column labeled Center, are the sums of all the probabilities.B 'If you get lost, press Home, End, and then the down arrow.6'Note that the X values increase UP the screen.6a'The values for X must lie between -11 and +11.H 'Be sure these sums are always 1. Otherwise, something is wrong.$u<7~7[\ 2=?=3@'SM230 - Spring 1997, Sums of Random Variables, J. Turner'Assignment:'1. Sums of binomials with the same p are binomial.@ '2. Sums of binomials with different p's are NOT binomial('3. The Central Limit Theorem (1) ` ` ` ` ` ` #` '` )`(,'4. The Central Limit Theorem (2)*2'5. Differences of random variablesZ'Show that for various values of N and p, if X and Y are binomial with the same p,P'then X+Y is binomial with the sum of the N's and the common value of p.2'For P(X) and P(Y), use @BINOMDIST(X,N,P,0)D'To the right of the blue column (X+Y), also put @BINOMDIST.;4'For this N, use the sum of the X and Y N's.PX 'It's easier if you put P in one cell and put the cell reference in the formula.h` 'Vary p and the N's and show that the blue column (X+Y) is always the same as @BINOMDISTT 'Do as #1, but use different values of p for X and for Y (and also for X+Y).vZ'Show that if the p's for X and Y are different, there is no value of p that makesf4'@BINOMIDIST for X+Y match the blue column.FV'It is easiest to take the p for X and the p for Y so that they average to 1/2,*'e.g., 0.3 and 0.7 or 0.1 and 0.9.:'Then, if any p would work for X+Y, it must be 1/2.D'Let the PMF for X and Y be 1/3 for X=-1,0,+1 and Y=-1,0,+1.=Ta'(We are not using BINOMDIST here. Just enter +1/3 in the cols for X and Y.)SP'Highlight the column of probabilities for X+Y. Initially, this is Col H.@a'Do this by clicking on the H near the top of the screen.2a'Next, select /Block,Values from the menu.Ha'Next, click on the column to the right of X+Y (Col I, probably),8a'once again, by clicking on the column letter.==a'Click on OK.D'This converts the formulas to values and stores the values.<B'With col. H still highlighted, select /Block,Values again.4a'This time select the green column (Col. F).eV 'This will transfer the probabilites for X+Y to Y and also re-calculate P(X+Y).\!'With the same (original) blue column still highlighted, select /Block,Values again.e"'Then click on Col. J.<#a'Select /Block,Values yet again and click on Col. F.4$'This will produce new probabilites for X+Y.e2%'Keep on repeating this process 6-10 times.`&a'Col. H should always remain highlighted. At each step, first copy the values to a columnB'a'to the right and then copy to Col. F. ORDER IS IMPORTANT.@('Graph all these sets of probabilities on the same axes.P)a'Highlight a rectangle of values starting in Col I and use a Line graph.>*'The last series should look like a bell-shaped curve.^-'Repeat the above process, but start with a different set of probabilites for X and Y.EH.'You may pick any range, but symmetric probabilities are nicer..H/'Generally, avoid zero probabilities - they make unpretty graphs.R0'The general shape of the resulting graph should be the same in all cases.>3'Suppose X has a binomial distribution with N=7, p=0.6:4'and Y has a binomial distribution with N=7, p=0.5.>5'What is the probability that Y will be larger than X?H6'The trick is to write down the probabilities for -X and use the7'same spreadsheet.H8'The probabilites for -X will go DOWN the sheet, like the others.V9a'Also, in BINOMDIST, use -X instead of X. Since X is negative, -X is positive.8:'Then we need to sum the probabilities for Y-X>0.2;'In an empty cell, type @SUM( and no Enter.:<'Then highlight the blue probabilities where Y-X>0.&='Then type ) and press Enter..dd Group14Group1KOOK*:MO*:OKCO:  # # #