Question

Let P be a probability distribution on sample space Ω and A ⊂ Ω an event such that P(A) > 0. Show that the conditional probability given A is a proper probability distribution on Ω using the axioms of probability and definition of conditional probability.

Answer #1

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...

Consider a probability space where the sample space is Ω = {
A,B,C,D,E,F } and the event space is 2 Ω . Assume that we only know
that the probability measure P {·} satisfies
P ( { A,B,C,D } ) = 4/5
P ( { C,D,E,F } ) = 4/5 .
a) If possible, determine P ( { D } ), or show that such a
probability cannot be determined unequivocally.
b) If possible, determine P ( {D,E,F } ),...

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 A1, . . . , An be a partition of the sample space Ω. Let X
be a random variable. What is the formula for E(X) in terms of the
conditional expected values E(X|A1), . . . , E(X|An)?
(b) Two coins are tossed. Then cards are drawn from a standard
deck, with replacement, until the number of “face” cards drawn (a
“face” card is a jack, queen, or king) equals the number of heads
tossed. Let X =...

Let A be an event, and let IA be the associated indicator random
variable: IA(ω)=1 if ω∈A, and IA(ω)=0 if ω∉A. Similarly, let IB be
the indicator of another event, B. Suppose that, P(A)=p, P(B)=q,
and P(A∪B)=r.
Find E[(IA−IB)2] in terms of p,q,r?
2.Determine Var(IA−IB) in terms of p,q,r?

Let A be an event, and
let IA be the associated indicator random variable: IA(ω)=1 if ω∈A,
and IA(ω)=0 if ω∉A. Similarly, let IB be the indicator of another
event, B. Suppose that, P(A)=p, P(B)=q, and P(A intersection
B)=r.
Find E[(IA−IB)2] in
terms of p,q,r?
2.Determine Var(IA−IB)
in terms of p,q,r?
The solution in Chegg
is for P(AUB)=r instead of P(A intersection B)=r. I need to know
how to find Var(IA−IB) in terms of p,q,r?

Suppose S is a sample space and f (E) = n(E) for each event E of
S. Prove that f is a probability
n(S) function by verifying that it obeys the three axioms.

Let A, B and C are defined as an event in a certain sample
space. The following information are known:
* A and C are independent events.
* A and B are mutually exclusive events.
* ?(? ∪ ?) = 0.99, ?(? ∪ ?) = 0.999, ?(?) = 0. 48
Find ?(?) and ?(?).

A probability experiment is conducted in which the sample space
of the experiment is S={7,8,9,10,11,12,13,14,15,16,17,18}, event
F={7,8,9,10,11,12}, and event G={11,12,13,14}. Assume that each
outcome is equally likely. List the outcomes in F or G. Find P(F or
G) by counting the number of outcomes in F or G. Determine P(F or
G) using the general addition rule.

For a discrete sample space the probability of an event is the
sum of all of the probabilities of the outcomes associated with the
event. true or false?

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