# Can I have some help for Statistics/probability theory

Let $$\Omega = \{0,1\}^{\mathbb{N}} = \{\alpha=(\alpha_1,\alpha_2,...):\alpha_i \in \{0,1\}\}$$

Fact. There exists a $$\sigma$$-algebra $$\mathcal{F}$$ such that for every $$\beta = (\beta_1,...,\beta_n)\in\{0,1\}^n$$, if we define the set $$E_\beta\subset\Omega$$ as

$$E_\beta=\{\alpha\in\Omega:(\alpha_1,\alpha_2,...,\alpha_n)=\beta\}$$, then $$E_\beta\in$$ $$\mathcal{F}$$.

For example, with $$\beta=(0,1,1,0), E_\beta$$ consists of all sequences $$(0,1,1,0,\alpha_5,\alpha_6,...)$$, where $$i\geq5, \alpha_i\in\{0,1\}$$.

$$P(E_\beta)=p^{\sum_{i=1}^n\beta_i}(1-p)^{n-\sum_{i=1}^n\beta_i}$$

Fact. $$P$$ can be extended to a probability measure on $$(\Omega,\mathcal{F})$$

Use below to show that if $$p>0$$ then $$P(\{0\})=0$$:

$$P\big(\bigcap_{n=1}^\infty B_n)=\lim_{n\to\infty}P(B_n)$$.

I am a little confused on where to start.

• Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. – gung - Reinstate Monica Apr 12 '19 at 14:16

I'm not entirely sure about your $$\Omega=\lbrace0, 1\rbrace^N$$, is $$\Omega$$ the set of all vectors of length exactly $$N$$ with entries in $$\lbrace 0, 1 \rbrace$$ or is $$N$$ misguiding and $$\Omega$$ is actually the set of all vectors of any length with entries in $$\lbrace 0, 1 \rbrace$$?
Anyway, this looks very much like a homework in measure theory/stochastics and as such I'd advise you to start with thinking about how to choose the events $$B_n$$ (in respect to a sequence of appropriate vectors $$\beta_n$$) such that you can derive the answer.
• Sorry It is the natural numbers $\mathbb{N}$ – i9-9980XE Apr 12 '19 at 14:00
• I told you how to start :) Anyway, if my hint confuses you, you need to study sequences/limits from your analysis classes and the definitions of probability measures and $\sigma$-algebras from your stochastics/measure theory class. – Edgar Apr 12 '19 at 14:12