if the density function of a bias coin is x-x^2 between 0-1. and z=0 is probability of bias coin fall on tail, z=1 is on its head, then whats probability of z=0,z=1.?

since the coin flip is discrete, and density function is continous, not sure how this problem would work out.

so maybe i should find F(Z) in term of x, then subsitude into the total probability?

Not sure if the question is specifying a \( \text{Beta}(2,2) \) as a prior distribution of the parameter \( p \) of the Bernoulli distribution - i.e. it is in a Bayesian setting, or so called "hierarchical model"

i kind figure out the above part
f(x)=x-x^2 is bias coin density function for probability of tail X from 0< x<1 else 0
so z(0) is probability of tail hence z(0)=integral(0,1) [z[z=0|x=0]*f(x) dx] z[z=0|x=0]=X

now if X^(z)=E[X|Z=z] is prediction of X whats X^[1].