Let A and B two events defined on a sample space . Show that A and...

Let A and B two events defined on a sample space . Show that A and B

are independent if and only if their characteristic functions 1A and 1B are independent random variables.

Expert Solution

TOPIC:Law of independence .

milcah answered 1 year ago

Let A and B be two events in a sample space S. If P(A1Bc) =.10, P(Ac1B)...

Let A and B be two events in a sample space S. If P(A1Bc) =.10, P(Ac1B) =.40 and P(Ac1Bc) =.10, a. Illustrate the events A1Bc, Ac1B and Ac1Bc on a van Venn-diagram b. Check for the independence of events A and B. c. Find P(A1B*AUB).

Let A and B be two subsets of the sample space of an experiment. If P(A)...

Let A and B be two subsets of the sample space of an experiment. If P(A) = 0.35, P(B) = 0.55, and P(A ∩ B) = 0.1, find (i) p(A ∩ Bc) (ii) p(A U B)c (iii) p(A ∩ B)c (iv) p(Ac ∩ Bc)

Let a < c < b, and let f be defined on [a,b]. Show that f...

Let a < c < b, and let f be defined on [a,b]. Show that f ∈ R[a,b] if and only if f ∈ R[a, c] and f ∈ R[c, b]. Moreover, Integral a,b f = integral a,c f + integral c,b f .

Two events ?1 and ?2 are defined on the same probability space such that: ?(?1 )...

Two events ?1 and ?2 are defined on the same probability space such that: ?(?1 ) = 0.7 , ?(?2 ) = 0.5 , and ?(?1 ??? ?2 ) = 0.3. a) Find ?(?1 ?? ?2 ). b) Find ?(?1 | ?2 ). c) Are ?1 and ?2 mutually exclusive (disjoint)? and why? d) Are ?1 and ?2 independent? and why?

Let A and B be two events in a sample with P(A)=0.4 and P(AuB)=0.7.Let P(B)=p a)i...

Let A and B be two events in a sample with P(A)=0.4 and P(AuB)=0.7.Let P(B)=p a)i For what value of p are A and B mutually exclusive? ii for what value of p are A and B independent? b) Assume that P(A)=0.4 and P(B)=0.3 i find P(B') ii if A and B are mutually exclusive , what is P(A or B) ii given that P(A or B)=0.6, Find the P(B/A) c) suppose events A and B are such that P(A)=0.25,P(B)...

Part V. 1. Events G and H are defined on the sample space. If P(G) =...

Part V. 1. Events G and H are defined on the sample space. If P(G) = 0.3, P(H) = 0.2, and P(G ! H) = 0.1 a) Are G and H independent? Explain. __________________________________________________________________________ b) Are G and H mutually exclusive? Explain. ___________________________________________________________________________ 2. The classical probability concept applies only when all possible outcomes are ______________________. 5 3. If A is the event that I win the lottery and B is the event that I go on vacation, then P(B...

Let A , B , and C be disjoint subsets of the sample space. For each...

Let A , B , and C be disjoint subsets of the sample space. For each one of the following statements, determine whether it is true or false. Note: "False" means "not guaranteed to be true." a) P(A)+P(Ac)+P(B)=P(A∪Ac∪B) b) P(A)+P(B)≤1 c) P(Ac)+P(B)≤1 d) P(A∪B∪C)≥P(A∪B) e) P((A∩B)∪(C∩Ac))≤P(A∪B∪C)P((A∩B)∪(C∩Ac))≤P(A∪B∪C) f) P(A∪B∪C)=P(A∩Cc)+P(C)+P(B∩Ac∩Cc) ) Please explain how you got the answer.

A fair coin is tossed two times, and the events A and B are defined as...

A fair coin is tossed two times, and the events A and B are defined as shown below. Complete parts a through d. A: {At most one tail is observed} B: {The number of tails observed is even} a. Identify the sample points in the events A, B, AunionB, Upper A Superscript c, and AintersectB. Identify the sample points in the event A. Choose the correct answer below. A. A:{TT comma HH} B. A:{TT} C. A:{HH comma HT comma TH}...

Let A, B and C be mutually independent events of a probability space (Ω, F, P),...

Let A, B and C be mutually independent events of a probability space (Ω, F, P), such that P(A) = P(B) = P(C) = 1 4 . Compute P((Ac ∩ Bc ) ∪ C). b) [4 points] Suppose that in a bicycle race, there are 19 professional cyclists, that are divided in a random manner into two groups. One group contains 10 people and the other group has 9 people. What is the probability that two particular people, let’s say...

1. Given events A and B from the same sample space and: P(A) = 0.2, P(B)...

1. Given events A and B from the same sample space and: P(A) = 0.2, P(B) = 0.6, P(A and B) = 0.1. Find the probability: P(A or B). 2. The scores of the class have a normal distribution with a mean of 77 and a standard deviation of 8. Find the 80th percentile score. a) 85 b) 0.375 c) 83.7 d) 84.8

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