Question

Bookmark this page Setup: All problems on this page will follow the definitions here: Let X, Y be two Bernoulli random variables and let P a r = = = P(X = 1) (the probability that X = 1) P(Y = 1) (the probability that Y = 1) P(X = 1, Y = 1) (the probability that both X = 1 and Y = 1). Let (X1,Y1), ... ,(Xn, Yn) be a sample of n i.i.d. copies of (X, Y). Based on this sample, we want to test whether X and Y are independent, i.e., whether r = pq.

Answer 1

Ang:- Guien P= p(x:1) q: p(y=1) Y : P(x-1, y-1) %3D Tn Calculalion :- let x,Y be ló Bernoil vrandon in and ib Y- Pg hen x and y are Veribte inokpendint (ie) p(x=1)(y:1) = P(x=1,Y=1) luo Bernaul® 1aribley X and P(X =1,4=D); %3D let A : Be the cuent lhat x=0, the euent that y=1, .. Pag proo : p(An1s')

2. PCANB') = P(A) - P(ANB) NOW KA) = P, P(AnB) =} * PCAns')- P-8 - P(x=1,y=0) -> P(x=0)(Y=1) pado is P(AnB) Reg P(B) - P(BNA) %3D NOw PlY=1) = P(B)- V Reg poroo 9-7 = P(x=0, Y =1) %3D » p(8:0, Y= 0) Fag prob planB') = P(AUB)' 7- P(AUB) demirgan law p(AhB')= P(AuB}

3) NOw PCANB') = |- p( AUB) = | - KA)- P(B) + P(aNB) = |-p-9+Y P(x=0,Y-0) = |- p-9+Y %3D ib A and B îndepenoent eent Hence then PCANB) = p(A) P(B) P(AnB) =Y; P(A)= P P(B)= q Hence then X andy are 26 8: Pa imdipendert P(x = 1) P(y=1) Say P(x=1,Y=1) anol %3B