expected value and variance

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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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@sonpham,

(b) 15/7

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@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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@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)?

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@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.

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@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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expected value and variance

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