# The Special Fish Report

## Albert W. Small (December 1944)

### Page 27 1/2

Tony Sale's
Codes and Ciphers

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CX/MSS
TOP SECRET Special Fish Report Page 27 1/2

It is a fact, however, that Sum |x1| is usually higher than Sum (x1 with regard

to signs of the true wheel) - and therefore the more nearly accurate

value of a pip (numerically) is:

R + qX
R - qX

wherein "q" is a factor for reducing X by the amout the maximum score

may be expected to exceed the true score. This is getting involved and

I hate to break up continuity here but it is really the best place

to thrash it out.

What value shall be given "q"?

We said that x = Sum|xi|. The expected value of |xi|

my be called Sum|xi|/W wherein W = wheel length, and therefore the

expected value of |xi| (or E|xi|) = X/W.

Also, qX = correct value = Sum (xiei) (wherein ei = +1 or -1

depending on the signs of the true wheel.) Let us call Sum (xiei)/W (the

expected value of xiei) = x0 and let us call sigma xiei = sigma

Now, since |xiei| is a function of xiei (whose expected value is x0)

we can derive the following directly from the normal function:

and we may graph x0 (which is actully Sigma xiei/W) against X/W Sigma and thereby

solve for x0 and obtain Sum(xiei) which is qX.

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