P(y) = exp(y*z)/SUM[y'=0..1](y'*z) = exp(y*z)/(exp(0*z)+exp(1*z)) = exp(y*z)/(1+exp(z)) 6.22

P(y) = Sigmoid((2*y-1)*z) 6.23

where

Sigmoid(x)= 1/(1+exp(-x))

In this chapter of the Deep Learning Book, y is a binary variable, which is either 0 or 1.

So:

P(y=0) = exp(0*z)/(1+exp(z)) = 1/(1+exp(z))

This means, the exponent x in the Sigmoid funcion corresponds to -z.

Furthermore:

P(y=1) = exp(1*z)/(1+exp(z))

Rephrasing:

exp(z)/(1+exp(z)) * exp(-z)/exp(-z) = exp(z-z)/(exp(...