2. 15 pts] Suppose E,, E. , En are independent events. Prove that に!
If A 1, . .., An are independent events, then n)- 1 _ ん に1
Prove that if A and B are independent events, then (a) A and B are independent. (b) A and Bc are independent.
Prove that (for two events A and B) if A and Bc are independent, then A and B are independent
Problem 1.25. Suppose you are given a sequence of events An, nEN that are independent and such that ΣηΕΝ P(An-oo. Prove that the event "An happens infinitely often" has probability one. This result is the reverse iel Note it nes the events to he independent.
Prove or disprove: Two events A and B are independent if and only if they are disjoint.
3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12. 3. (15 pts.) Let A e Rmxn be a full rank matrix, m > n. Suppose that Let r = Ax-b. Prove that reprthogonal to Az minimizes llAz-b12.
Suppose that events E and F are independent, P(E) 0.3, and P(F) 0.8. What is the P(E and F)? The probability P(E and F) is (Type an integer or a decimal.)
Suppose E and F are independent events. Find Pr[E′∩F] if Pr[E]=1/3 and Pr[F]=1/3 A and B are independent events. If Pr(A∩B)=0.24 and Pr[A]=0.3, what is Pr[B]?
3. Suppose E, and Eare independent. Either give a counter example or demonstrate that P(En E2|F) = P(E|F)P(E2|F).
Problem VI.(15 pts.) Suppose that is an irrational number. 1. Prove that j + cannot be a rational number 9 with gl < 2. 2. Can j + be a rational number whose absolute value is greater than 2? Why or why not?