Let (Ω, F , P) be a probability space. Suppose that Ω is the collection of...

Let (Ω, F , P) be a probability space. Suppose that Ω is the collection of all possible outcomes of a single iteration of a certain experiment. Also suppose that, for each C ∈ F, the probability that the outcome of this experiment is contained in C is P(C).
Consider events A, B ∈ F with P(A) + P(B) > 0. Suppose that the experiment is iterated indefinitely, with each iteration identical and independent of all the other iterations, until it results in an outcome that is an element of A ∪ B, after which it stops. What is the probability that this procedure results in an outcome that is an element of A? Do not use conditional probability to answer this question.

Expert Solution

orchestra answered 1 year ago

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...

Exercise 4 (Indicator variables). Let (Ω, P) be a probability space and let E ⊆ Ω...

Exercise 4 (Indicator variables). Let (Ω, P) be a probability space and let E ⊆ Ω be an event. The indicator variable of the event E, which is denoted by 1E , is the RV such that 1E (ω) = 1 if ω ∈ E and 1E(ω)=0ifω∈Ec.Showthat1E isaBernoullivariablewithsuccessprobabilityp=P(E). Exercise 5 (Variance as minimum MSE). Let X be a RV. Let xˆ ∈ R be a number, which we consider as a ‘guess’ (or ‘estimator’ in Statistics) of X . Let...

Let Prob be a probability function on the power set P(Ω) of a sample space Ω....

Let Prob be a probability function on the power set P(Ω) of a sample space Ω. Let B be a set such that Prob(B) > 0. For any set A, define G(A|B) = Prob(A ∩ B) Prob(B) . Prove that for fixed B and as as a function of A, G(·|B) is also a probability function, meaning that G(·|B) satisfies the axioms of probability..

Let Ω be any set and let F be the collection of all subsets of Ω...

Let Ω be any set and let F be the collection of all subsets of Ω that are either countable or have a countable complement. (Recall that a set is countable if it is either finite or can be placed in one-to-one correspondence with the natural numbers N = {1, 2, . . .}.) (a) Show that F is a σ-algebra. (b) Show that the set function given by μ(E)= 0 if E is countable ; μ(E) = ∞ otherwise...

Let (?,?,?) be a probability space and suppose that ?∈? is an event with ?(?)>0. Prove...

Let (?,?,?) be a probability space and suppose that ?∈? is an event with ?(?)>0. Prove that the function ?:?→[0,1] defined by ?(?)=?(?|?) is a probability on (?,?).

Let ? be the sample space of an experiment and let ℱ be a collection of...

Let ? be the sample space of an experiment and let ℱ be a collection of subsets of ?. a) What properties must ℱ have if we are to construct a probability measure on (?,ℱ)? b) Assume ℱ has the properties in part (a). Let ? be a function that maps the elements of ℱ onto ℝ such that i) ?(?) ≥ 0 , ∀ ? ∈ ℱ ii) ?(?) = 1 and iii) If ?1 , ?2 … are...

(3) Let V be a vector space over a field F. Suppose that a ? F,...

(3) Let V be a vector space over a field F. Suppose that a ? F, v ? V and av = 0. Prove that a = 0 or v = 0. (4) Prove that for any field F, F is a vector space over F. (5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1, a2, a3 ? R} of polynomials of degree ? 3 is a vector space over...

Let X be a compact space and let Y be a Hausdorff space. Let f ∶...

Let X be a compact space and let Y be a Hausdorff space. Let f ∶ X → Y be continuous. Show that the image of any closed set in X under f must also be closed in Y .

Let T∈ L(V), and let p ∈ P(F) be a polynomial. Show that if p(λ) is...

Let T∈ L(V), and let p ∈ P(F) be a polynomial. Show that if p(λ) is an eigenvalue of p(T), then λ is an eigenvalue of T. Under the additional assumption that V is a complex vector space, and conclude that {μ | λ an eigenvalue of p(T)} = {p(λ) | λan eigenvalue of T}.

Suppose E and F are two mutually exclusive events in a sample space S with P(E)...

Suppose E and F are two mutually exclusive events in a sample space S with P(E) = 0.34 and P(F) = 0.46. Find the following probabilities. P(E ∪ F) P(EC) P(E ∩ F) P((E ∪ F)C) P(EC ∪ FC)

Question