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The following simple random variable is in canonical form:

X = - 3 . 75 I A - 1 . 13 I B + 0 I C + 2 . 6 I D .

Express the events { X ( - 4 , 2 ] } , { X ( 0 , 3 ] } , { X ( - , 1 ] } , { | X - 1 | 1 } , and { X 0 } in terms of A , B , C , and D .

  • A B C
  • D
  • A B C
  • C
  • C D

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Random variable X , in canonical form, is given by X = - 2 I A - I B + I C + 2 I D + 5 I E .

Express the events { X [ 2 , 3 ) } , { X 0 } , { X < 0 } , { | X - 2 | 3 } , and { X 2 4 } , in terms of A , B , C , D , and E .

  • D
  • A B
  • A B
  • B C D E
  • A D E

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The class { C j : 1 j 10 } is a partition. Random variable X has values { 1 , 3 , 2 , 3 , 4 , 2 , 1 , 3 , 5 , 2 } on C 1 through C 10 , respectively. Express X in canonical form.

T = [1 3 2 3 4 2 1 3 5 2];[X,I] = sort(T)X = 1 1 2 2 2 3 3 3 4 5 I = 1 7 3 6 10 2 4 8 5 9
X = I A + 2 I B + 3 I C + 4 I D + 5 I E
A = C 1 C 7 , B = C 3 C 6 C 10 , C = C 2 C 4 C 8 , D = C 5 , E = C 9
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The class { C j : 1 j 10 } in [link] has respective probabilities 0.08, 0.13, 0.06, 0.09, 0.14, 0.11, 0.12, 0.07, 0.11, 0.09. Determinethe distribution for X .

T = [1 3 2 3 4 2 1 3 5 2];pc = 0.01*[8 13 6 9 14 11 12 7 11 9];[X,PX] = csort(T,pc);disp([X;PX]')1.0000 0.2000 2.0000 0.26003.0000 0.2900 4.0000 0.14005.0000 0.1100
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A wheel is spun yielding on an equally likely basis the integers 1 through 10. Let C i be the event the wheel stops at i , 1 i 10 . Each P ( C i ) = 0 . 1 . If the numbers 1, 4, or 7 turn up, the player loses ten dollars; if the numbers 2, 5,or 8 turn up, the player gains nothing; if the numbers 3, 6, or 9 turn up, the player gains ten dollars; if the number 10 turns up, the player loses one dollar. The randomvariable expressing the results may be expressed in primitive form as

X = - 10 I C 1 + 0 I C 2 + 10 I C 3 - 10 I C 4 + 0 I C 5 + 10 I C 6 - 10 I C 7 + 0 I C 8 + 10 I C 9 - I C 10
  • Determine the distribution for X , (a) by hand, (b) using MATLAB.
  • Determine P ( X < 0 ) , P ( X > 0 ) .
p = 0.1*ones(1,10); c = [-10 0 10 -10 0 10 -10 0 10 -1]; [X,PX]= csort(c,p); disp([X;PX]') -10.0000 0.3000-1.0000 0.1000 0 0.300010.0000 0.3000 Pneg = (X<0)*PX' Pneg = 0.4000Ppos = (X>0)*PX' Ppos = 0.300
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A store has eight items for sale. The prices are $3.50, $5.00, $3.50, $7.50, $5.00, $5.00, $3.50, and $7.50, respectively.A customer comes in. She purchases one of the items with probabilities 0.10, 0.15, 0.15, 0.20, 0.10 0.05, 0.10 0.15. Therandom variable expressing the amount of her purchase may be written

X = 3 . 5 I C 1 + 5 . 0 I C 2 + 3 . 5 I C 3 + 7 . 5 I C 4 + 5 . 0 I C 5 + 5 . 0 I C 6 + 3 . 5 I C 7 + 7 . 5 I C 8

Determine the distribution for X (a) by hand, (b) using MATLAB.

p = 0.01*[10 15 15 20 10 5 10 15];c = [3.5 5 3.5 7.5 5 5 3.5 7.5];[X,PX] = csort(c,p);disp([X;PX]')3.5000 0.3500 5.0000 0.30007.5000 0.3500
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Suppose X , Y in canonical form are

X = 2 I A 1 + 3 I A 2 + 5 I A 3 Y = I B 1 + 2 I B 2 + 3 I B 3

The P ( A i ) are 0.3, 0.6, 0.1, respectively, and the P ( B j ) are 0.2 0.6 0.2. Each pair { A i , B j } is independent. Consider the random variable Z = X + Y . Then Z = 2 + 1 on A 1 B 1 , Z = 3 + 3 on A 2 B 3 , etc. Determine the value of Z on each A i B j and determine the corresponding P ( A i B j ) . From this, determine the distribution for Z .

A = [2 3 5];B = [1 2 3];a = rowcopy(A,3); b = colcopy(B,3);Z =a + b % Possible values of sum Z = X + Y Z = 3 4 64 5 7 5 6 8PA = [0.3 0.6 0.1];PB = [0.2 0.6 0.2];pa= rowcopy(PA,3); pb = colcopy(PB,3);P = pa.*pb % Probabilities for various values P = 0.0600 0.1200 0.02000.1800 0.3600 0.0600 0.0600 0.1200 0.0200 [Z,PZ]= csort(Z,P); disp([Z;PZ]') % Distribution for Z = X + Y 3.0000 0.06004.0000 0.3000 5.0000 0.42006.0000 0.1400 7.0000 0.06008.0000 0.0200
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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