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 (?,?).

Solutions

Expert Solution

TOPIC:Probability measure.

orchestra answered 1 year ago

Next > < Previous

Related Solutions

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...

Suppose my sample space consists of current STATS125 students. Let T be the event that a...

Suppose my sample space consists of current STATS125 students. Let T be the event that a randomly selected current STATS125 student usually attends Tuesday lectures, W be the event that a randomly selected current STATS125 student usually attends Wednesday lectures H be the event that a randomly selected current STATS125 student usually attends Thursday lectures. a) Draw 4 Venn diagrams and use shading to identify the following sets: i. T ? H ii. T^complement ? (W ? H) iii. the...

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic...

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

Let Bt be a 1-dimensional Brownian motion and let c > 0 be a constant. Prove...

Let Bt be a 1-dimensional Brownian motion and let c > 0 be a constant. Prove that is also a Brownian motion.

hzhshs assume that X is normed linear space let A is a subset of X prove...

hzhshs assume that X is normed linear space let A is a subset of X prove that: if A is compact then A is closed and bounded

Let X and Y be T2-space. Prove that X*Y is also T2

#18 Events A and B are mutually exclusive. Suppose event occurs with probability 0.05 and event...

#18 Events A and B are mutually exclusive. Suppose event occurs with probability 0.05 and event occurs with probability 0.36. a. Compute the probability that B occurs or does not occur (or both). b. Compute the probability that either occurs without occurring or occurs without occurring. (If necessary, consult a list of formulas.) ? #19 Events and are mutually exclusive . Suppose event occurs with probability 0.11 and event B occurs with probability 0.81. If does not occur, what is...

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...

1. The probability of event A is 60%, the probability of event B is 40%, and...

1. The probability of event A is 60%, the probability of event B is 40%, and the probability of either A or B is 65%. What is the probability of events A and B simutaneously? 2. 50% of items are Type A, 30% are Type B, and 20% are Type C. each type is broken up into V1 and V2. Type A is 80% V1, Type B is 60% V1, and Type C is 30% V1. If a randomly selected...

Prove the following statements! 1. Let S = {0, 1, . . . , 23} and...

Prove the following statements! 1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by (a) g(m) = f(7m) then g is injective and (b) g(m) = f(15m) then g is not injective. 2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective. 3. Let f : A→B and g : B→C be surjective....

Subjects