#C6.txt:Binomial distributions
Help:=proc(): print(` Bp(n,p,k), HT(n,p,x) , dbeta(p,a,b)`): end:

#Bp(n,p,k): The prob. mass function of the Binomial distribution, tossing a coin n times
#Pr(H)=p, k=0..n, Pr(X=k)

Bp:=proc(n,p,k)  binomial(n,k)*p^k*(1-p)^(n-k): end:

#HT(n,p,x): Frequentist Hypothesis testing. If the owner of the coin claims that Pr(H) is at least
#p what is the cutoff of number of Heads below which you can be confident with level of confidence
#x that he is a lier.
HT:=proc(n,p,x) local k1,k,su:

for k from 0 to n while add(Bp(n,p,k1),k1=0..k)<=1-x do od:
k-1:
end:

dbeta:=proc(p,a,b): GAMMA(a+b)/(GAMMA(a)*GAMMA(b))*p^(a-1)*(1-p)^(b-1):end:

#Post~Likelihood*Prior 	~ p^k*(1-p)^(n-k)*dbeta(a,b)=p^k*(1-p)^(n-k)*p^(a-1)*(1-p)^(b-1)=
#p^(k+a-1,n-k+b-1)= dbeta(k+a,n-k+b)