Probability Axioms

**n7615r** · May 15th 2009, 07:15 AM

How would you go about doing this? is it really obvious?

**Isomorphism** · May 15th 2009, 07:53 AM

Originally Posted by n7615r

How would you go about doing this? is it really obvious?

So where are you having trouble? Is it in:

The first part : Reading set notations(about the infinitely often)

The second part : The union bound axiom? the one that says probability of union of countable number of events is upperbounded by the sum of probability of those events

In case of basic application problems like these, its beneficial to both (us and you), if you tell us where you are having trouble.

**n7615r** · May 15th 2009, 08:22 AM

i'm having trouble understanding the set notation and how 'infintely often can be written in that way.

and also showing how the P(A)≤P(Bn). is it because A is a subset of Bn?

and how does this relate to the third part?

**Moo** · May 15th 2009, 10:33 AM

Hello,

If $B_n$ happens, it means that there is at least one $A_m$ in its union that happens.
This can be translated as " $\exists m\geq n$ , such that $A_m$ happens"

Then, for $A=\bigcap_{n\geq 1} B_n$ to happen, $B_n$ has to happen for any $n\geq 1$

This can finally be translated as "for any n, there exists m>n such that $A_m$ happens".
In other words, there are infinitely many $A_m$ events that happen.

You can read this for more things : http://www.mathhelpforum.com/math-he...up-liminf.html in particular post #4, by Laurent

and also showing how the P(A)≤P(Bn). is it because A is a subset of Bn?

Probability axioms - Wikipedia, the free encyclopedia
Since A is, as you noticed, a subset of $B_n$ , then there exists a set $E$ , disjoint from A, such that $A\cup E=B_n$
By the third axiom, we have (the red equality) :
$\mathbb{P}(B_n)=\mathbb{P}(A\cup E){\color{red}=}\mathbb{P}(A)+\mathbb{P}(E)$

Now, by the first axiom, we know that $\mathbb{P}(E)\geq 0$

And it follows that $\mathbb{P}(B_n)\geq \mathbb{P}(A)$

and how does this relate to the third part?

Note that P( $A_n$ happens infinitely often)=0 $\Leftrightarrow$ P(A)=0

And note that if $\sum_{m\geq 1} a_m<\infty$ , this implies that the partial reminder (I don't know the exact name) $\sum_{m\geq n} a_m \xrightarrow[]{n\to\infty} 0$
It should be okay from now

The second part : The union bound axiom? the one that says probability of union of countable number of events is upperbounded by the sum of probability of those events

Hmm this axiom isn't stated in the probability axioms. Nor it is stated in the axioms defining a measure. Though it's a direct consequence. I know the French name, but not sure in English... I think it is called sigma subadditivity or something like that...

**n7615r** · May 15th 2009, 10:43 AM

Thanks so much! this was a great help

**Isomorphism** · May 15th 2009, 11:45 AM

Originally Posted by Moo

Hmm this axiom isn't stated in the probability axioms. Nor it is stated in the axioms defining a measure. Though it's a direct consequence. I know the French name, but not sure in English... I think it is called sigma subadditivity or something like that...

I know that, but since its equivalent to the axiom, I generally find it convenient to remember in that form

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