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Guy McSucker
12-15-2004, 10:41 AM
Via a roundabout route, I've been looking at this post (http://archiveserver.twoplustwo.com/showthreaded.php?Cat=&amp;Number=683150&amp;page=&amp;view=&amp;sb =5&amp;o=&amp;vc=1) by BruceZ containing his own derivation of a risk of ruin formula, using coin-flip games as the basis.

The game is an even-money bet of size b with a biased coin: you win \$b with probability p and lose \$b with probability q = 1-p.

BruceZ writes this formula for the variance of the game:

variance = p*b^2 + q*(-b)^2  EV^2

Is that right? I don't know much stats, but I thought variance was computed as the average of the squares of the deviations of actual results from the mean, i.e.

p(b - EV)^2 + q(-b - EV)^2

which doesn't seem to be the same thing.

On the other hand, BruceZ is pretty reliable in these matters.

Can anyone shed any light on this for me?

Guy.

BruceZ
12-15-2004, 12:16 PM
[ QUOTE ]
Via a roundabout route, I've been looking at this post (http://archiveserver.twoplustwo.com/showthreaded.php?Cat=&amp;Number=683150&amp;page=&amp;view=&amp;sb =5&amp;o=&amp;vc=1) by BruceZ containing his own derivation of a risk of ruin formula, using coin-flip games as the basis.

The game is an even-money bet of size b with a biased coin: you win \$b with probability p and lose \$b with probability q = 1-p.

BruceZ writes this formula for the variance of the game:

variance = p*b^2 + q*(-b)^2  EV^2

Is that right? I don't know much stats, but I thought variance was computed as the average of the squares of the deviations of actual results from the mean, i.e.

p(b - EV)^2 + q(-b - EV)^2

which doesn't seem to be the same thing.
Guy.

[/ QUOTE ]

Sure it is. Expand it out.

pb^2 + q(-b)^2 + EV^2(p + q) - 2bEV(p - q)

p + q = 1
EV = pb + q(-b) = b(p - q)
-2bEV(p - q) = -2EV^2

so we have

pb^2 + q(-b)^2 + EV^2 - 2EV^2

= pb^2 + q(-b)^2 - EV^2

Actually this form is just

variance = E(x^2) - [E(x)]^2

where E means expected value.

Guy McSucker
12-15-2004, 12:37 PM
[ QUOTE ]

EV = pb + q(-b) = b(p - q)

[/ QUOTE ]

Okay, that's the simplification I missed.

Thanks very much!

Feeling stupid,

Guy.