Let, X_1, X_2...X_N. be i.i.d from the uniform distribution on (0,1), and let M=max{X_1, X_2.X_N}, show that X_1/M and M are independent

I know probably I need to show X_1/M is minimal sufficient and M is ancillary, and use Basu theorem but I don't know how to do it.


Let (X_1,Y_1),(X_2,Y_2) (X_N,Y_N) be i.i.d. and absolutely continuous with common density

f(x,y)= 2/ theta^2 , x>0, y>0, x+y<theta
f(x,y)= 0 , otherwise

(This is the density for a uniform distribution on the region inside a triangle in R^2 )

a)find a minimal sufficient statistic for the family of joint distributions.
b)Find the density for your minimal sufficient statistic
c)Is the minimal sufficient statistic complete.

My answer is T=max{(X_1,Y_1),(X_2,Y_2) (X_N,Y_N)} is minimal sufficient but I don't know if it is correct, also the rest .


Let , X_1,X_2.X_N be i.i.d from a discrete distribution Q on {1,2,3}. Let p_i=Q({i})=P(X_j=i), i=1,.3. and assume we know that p_1=1/3, but have no additional knowledge of Q. Define

N_i= # { j <=n: X_j=i }

a)show that T =(N_1, N_2 ) is sufficient
b)Is T minimal sufficient ? If so, explain why. If not , find a minimal sufficient statistic.
I don't know how to work part b)

Thanks a bunch!