1. Consider two independent events, A and B, where 0< P(A) <1,0< P(B)< 1. Prove that...
Consider two independent events, A and B, where 0くP(A) < 1,0くP(8)く1. Prove that A' and B' are independent as well.
3. Let (X1, X2) have the joint p.d.f 1 if 0 <1,0 < <1 f(1, ) else Find P(X1X2 < 0.5)
B] <P[AP[B], and of two events A and B Give an example of two events such that P[ A with P[An B] > P[A]P[B].
Suppose two events A and B are two independent events with P(A) > P(B) and P(A U B) = 0.626 and PA กั B) 0.144, determine the values of P(A) and P(B).
4. Suppose A, B, C are events such that P(A), P(B), P(C) a. If (A, B, C) are independent, show that P(AU BUC)- b. If A, B, C are only pairwise independent, show that 17 24 SHA UBUC)<19 24
Suppose that A1,A2,.., Ak are mutually exclusive events and P(B)>0. Prove that
1. Use the formula P(A) PABP(B) + P(AlBc)P(B") to prove that if P(AB) P (AlBc) then A and B are independent. Then prove the converse (that if A and B are independent then P(AIB)- P(ABe). [Assume that P(B) > 0 and P(B) > 0.]
is independent of X, and e Problem 3 Suppose X N(0, 1 -2) -1 <p< 1. (1) Explain that the conditional distribution [Y|X = x] ~N(px, 1 - p2) (2) Calculate the joint density f(x, y) (3) Calculate E(Y) and Var(Y) (4) Calculate Cov(X, Y) N(0, 1), and Y = pX + €, where
Suppose that f (x II 2y), 0 < x < 1,0 < y < 1. Find EX + Y).
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?