[Hello,

I would like to know if someone can help for the question as below?

Suppose that P(E) is 0 or 1, show that E is independent of all events A.

Suppose that E is independent of itself, show that P(E) is 0 or 1.

First note that . Hence,
if , then

Then using the inclusion-exclusion principle, consider

If , then

But again we know that , hence

is independent from itself if and only if

Result follows.

Off the topic,but can you tell me how do you use all those symbols?
I use word and I could never find sqrt sign :|
Also integral.

Oh its an html code perhaps, I clicked and got the tags.:P

Hi Sadie. I think the answer you are after is in the General Discussions, under Math Typesetting in LaTex.

Thanks a lot, it's very helpful

BGM

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Suppose that (Ω ,F, P) is a probability space and E belong F satisfies p(E)>0 .
Let Q: F ----> [0,1] be defined by Q(A)=P(A/E) . Show that (Ω ,F, Q) is a probability space

To check is a probability space,
it is sufficient to check is a probability measure,
by checking the three axioms of probability.

http://en.wikipedia.org/wiki/Probability_axioms

Since is a valid probability measure, it satisfy
all 3 axioms of probability.

1. Note as

By the axiom of probability,

2. Note
Hence,

3. Note

Hence,

(by the axiom of probability)

Q.E.D.

BGM, you are a machine.

Thank you very much wish you a good life