22D Page 50
(d) Frequency of letters in Ψ'.
The number of dots and crosses in each impulse of the Ψ' stream are equal and their positions relatively independent. Therefore the frequency of every letter in the Ψ' stream is approximately equal.
(e) Frequency of letters in ΔΨ'.
In the ΔΨ' stream, though there are an equal number of dots and crosses in each impulse, they are so placed that there is a dot in every impulse at each extension.
T.M. dot positions occur (1-a) of the time and at each of these there is a stroke in ΔΨ'.
The ΔΨ' stream at T.M. cross positions is in fact the ΔΨ stream (unextended) and the chance of a cross in any impulse is b. Therefore the frequency of various letters is as follows
|/||0 crosses||(1-a) + a (1-b)5|
|9T34E||1 cross||ab (1-b)4|
|HONRLIADZS||2 crosses||ab2 (1-b)3|
|MOGPUWJFBY||3 crosses||ab3 (1-b)2|
|VQ5KX||4 crosses||ab4 (1-b)|
|ΔΨ'ij = dot, when TM = dot|
|When TM = cross
||P(ΔΨij = dot) = b2 + (1-b)2
= 2b2 - 2b + 1
|∴||ΔΨ'ij. with probability|| (1-a) + a(2b2 -2b + 1)
= 1 - a + b - 1 + a = b
(g) ΔΨ' stream and limitation.
In each impulse of ΔΨ' stream and in limitation stream there are an equal number of dots and crosses.
|Now at TM dot positions,||ΔΨ'i = dot|
|L = cross.|
Therefore the remaining ΔΨ'i dots, and the remaining lim. x's form the same proportion of the TM cross positions.
|Therefore at TM cross positions||ΔΨ'i cross|
|Consequently in any position,|
|P(ΔΨ'i = dot) = P(L+x = dot)|