Question

Let S be the sample space of an experiment and let ℱ be a non-empty collection...

Let S be the sample space of an experiment and let ℱ be a non-empty collection of subsets of S such that i) ? ∈ ℱ ⇒ ? ′ ∈ ℱ and ii) ?1 ∈ ℱ and ?2 ∈ ℱ ⇒ ?1 ∪ ?2 ∈ ℱ

a) Show that if ?1 ∈ ℱ and ?2 ∈ ℱ then ?1 ∩ ?2 ∈ ℱ .

b) Show that ? ∈ ℱ.

c) Is ℱ necessarily a ?-algebra? Explain briefly. A rigorous proof is not necessary.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let S be a collection of subsets of [n] such that any two subsets in S...
Let S be a collection of subsets of [n] such that any two subsets in S have a non-empty intersection. Show that |S| ≤ 2^(n−1).
Let f : X → Y and suppose that {Ai}i∈I is an indexed collection of subsets...
Let f : X → Y and suppose that {Ai}i∈I is an indexed collection of subsets of X. Show that f[∩i∈IAi ] ⊆ ∩i∈I f[Ai ]. Give an example, using two sets A1 and A2, to show that it’s possible for the LHS to be empty while the RHS is non-empty.
Let the S = {0,1,2,3,4,5,6,7,8,9,10} be the sample space of a random experiment. Suppose A is...
Let the S = {0,1,2,3,4,5,6,7,8,9,10} be the sample space of a random experiment. Suppose A is the event that we observe a number less than 3 and B be the event that we observe a number greater than 8. Determine the event that either A occurs or B occurs. Group of answer choices {3,4,5,6,7,8} {0,1,2,3,8,9,10} empty set {0,1,2,9,10} {4,5,6,7}
Let A, B be non-empty subsets of R. Define A + B = {a + b...
Let A, B be non-empty subsets of R. Define A + B = {a + b | a ∈ A and b ∈ B}. (a) If A = (−1, 2] and B = [1, 4], what is A + B?
Proposition 16.4 Let S be a non–empty finite set. (a) There is a unique n 2...
Proposition 16.4 Let S be a non–empty finite set. (a) There is a unique n 2 N1 such that there is a 1–1 correspondence from {1, 2,...,n} to S. We write |S| = n. Also, we write |;| = 0. (b) If B is a set and f : B ! S is a 1–1 correspondence, then B is finite and |B| = |S|. (c) If T is a proper subset of S, then T is finite and |T| <...
A Bernoulli trial is an experiment with the sample space S = {success, failure}. Let X...
A Bernoulli trial is an experiment with the sample space S = {success, failure}. Let X be the random variable on S defined by ? X(s) : 1 if s = success 0 if s = failure. Suppose the probability P ({success}) is equal to some value p ∈ (0, 1). (a) Tabulate for the rv X the probability distribution Px (X = x) in terms of p for x ∈ {0, 1}. (b) Give the expression for the cdf...
Let A, B be events of a sample space S with probabilities P(A) = 0.25, P(B)...
Let A, B be events of a sample space S with probabilities P(A) = 0.25, P(B) = 0.35 and P(A∪B) = 0.4. Calculate (i) P(A|B) ,(ii) P(B|A), (iii) P(A∩B), (iv) P(A|B).
Let S be a sample space with probability P and let A ⊂ S, B ⊂...
Let S be a sample space with probability P and let A ⊂ S, B ⊂ S be independent events. Given P (B) = 0.3 and P (A ∪ B) = 0.65, find P (A).
Let A, B and C be events in a sample space, S. Suppose that S =...
Let A, B and C be events in a sample space, S. Suppose that S = A ∪ B ∪ C, and that the following hold: A ∩ C = ∅, B ∩ C = ∅. Let P(A) = 0.3, P(B) = 0.5 and P(C) = 0.5. Find P(A ∩ B).
Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with the...
Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with the experiment having the following probability distribution. (Enter your answers as fractions.) Outcome s1 s2 s3 s4 s5 s6 Probability 3 12 1 12 4 12 1 12 2 12 1 12 (a) Find the probability of A = {s1, s3}. (b) Find the probability of B = {s2, s4, s5, s6}. (c) Find the probability of C = S.