B] <P[AP[B], and of two events A and B Give an example of two events such...
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).
Suppose that A1,A2,.., Ak are mutually exclusive events and P(B)>0. Prove that
Show that this grammar is ambiguous for the string a+b+c: <S> - <x> <X> - <x>+ <x> <X> - <id> <id> - abc Give the derivations.
1. Consider two independent events, A and B, where 0< P(A) <1,0< P(B)< 1. Prove that A and B' are independent as well.
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.
Exercise 6.14 Let y be distributed Bernoulli P(y = 1) unknown 0<p<1 p and P(y = 0) = 1-p f or Some (a) Show that p E( (b) Write down the natural moment estimator p of . (c) Find var (p) (d) Find the asymptotic distribution of vn (-p) as no. as n> OO.
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.]
3. Suppose that A and B are two events defined over the same sample space, with probabilities P(A) 3/4 and P(B)- 3/8. (a) Show that P(A UB) 2 3/4. (b) Show that 1/8 < P(AB) 3/8 (c) Give inequalities analogous to (a) and (b) for P(A) 2/3 and P(B)1/2.
(5) Use induction to show that Ig(n) <n for all n > 1.
0.2 Question 7 (1 point) <Venn 3> There are 2 events: A, B with P(A)-0.5, P(B)-0.4, P(AUB)-0.7 Find P(BA) (2 decimal places without rounding-up) Question 8 (1 point) Saved <Venn 4>