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expected value and variance

 
 
sonpham
 
Reply Sun 9 Oct, 2011 10:08 am
a games is played on a board marked off into a line of four squares. a player starts on the square numbered 2. on each turn, a six-sided die is thrown, and the players moves one square to the left if the die comes up 5 or 6, and one square to the right otherwise. the game ands when the player reaches either and of the board (square 1 or square 4).

[1][2][3][4]

a/ let x is the numbers of moves until the game ends. find the frequency function of x
b/ find E(x)
c/ find Var(x).
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markr
 
  1  
Reply Sun 9 Oct, 2011 10:08 pm
@sonpham,
(b) 15/7
sonpham
 
  1  
Reply Mon 10 Oct, 2011 07:46 am
@markr,
how did you get 15/7? is it the sum of x.p(x) from x=1,2,3,4,5,6
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engineer
 
  1  
Reply Mon 10 Oct, 2011 08:14 am
@markr,
Ok, I'm impressed. 15/7 is correct, but how did you calculate it?

If x is even, p(x) = (4/9) (2/9)^[(x-2)/2]
If x is odd, p(x) = (1/3) (2/9)^[(x-1)/2]

But given that, how did you get the sum of the quasi geometric series x.p(x)?
engineer
 
  1  
Reply Mon 10 Oct, 2011 09:01 am
@engineer,
Ok, I finally figured out how to do the infinite series, but I must have done it the hard way. If I posted my solution, it would cover a page.
0 Replies
 
markr
 
  1  
Reply Mon 10 Oct, 2011 09:22 pm
@engineer,
Code:
X B C X

1/3
B-------------->X
|\ ^
| \ |
| \ 1/3 |2/3
| ---<------- |
| \|
--------------->C
2/3

Expected number of turns to get to X from B or C:

B = 2/3(C+1) + 1/3
C = 1/3(B+1) + 2/3

B = 15/7
C = 12/7



I merely applied a technique I encountered several years ago. Figuring out the infinite series is much more impressive. Is there a documented technique you can point us to?

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