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I'm reading material about Markov chain and confused with notation of probabilities (probably something more than notation). Could anyone explain the difference between these notations, please.

Given probability space $(\Omega,\mathcal{F},\Pr)$, where $\Omega$ is a sample space, $\mathcal{F}$ is an event set and $\Pr$ is a probability assignment function. Let $X$ be a random variable in this space:

1.Does this mean random variable $X$ is in $\Omega$? But X is a function $X:\Omega\to S$ where $S$ is a state space.

2.If I write $X=x$, then $x\in\mathcal{F}$, is this right?

3.How does $\Pr(X)$ differ from $\Pr(X=x)$ and $\Pr(x)$?

4.Is a realization $x$ of $X$ a fixed real number?

5.Does $\Pr(X=x)$ means $\Pr(X(\omega)=x),\omega\in \Omega$?

Thanks in advance.

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  • $\begingroup$ 1 $X$ is a function with domain $\Omega$ and range $S \subset R$. 2. $x \in S$. 3. The second notation is clear, the other two are not. 4. Yes. 5 If you denote the event $A=\{ \omega: X(\omega)=x\} \in F$ then $Pr(X=x) = Pr (A)$. $\endgroup$ – kmitov Oct 26 '15 at 6:46
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  1. Random variable is just a function $X: \omega\mapsto X(\omega)$ (namely Borel function).
  2. If we write $X=x$ it could be $X(\omega)=x$ for some $\omega$ or $X=x$ for all $\omega$ (constant variable). Here $x\in S$.
  3. $Pr(X)$, $Pr(x)$ in general do not make sense. But $Pr(X=x)= Pr(\omega\mid X(\omega) = x)$ --- is the probability of the event $\{X(\omega) = x\}$.
  4. Usually $x$ is fixed number ($x\in S$).
  5. As in 3.

Hope this helps.

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  • $\begingroup$ Thanks for your and other replies. Lots of papers just write $\Pr(X)$ and I'm really confused with that. $\endgroup$ – Lvergergsk Oct 26 '15 at 7:01
  • $\begingroup$ @Lvergergsk Well, it could depends on context. Would you show an example? $\endgroup$ – Nikita Evseev Oct 26 '15 at 7:05
  • $\begingroup$ Yes, thanks for your help. For example it says $\Pr(O|Q,\lambda)=\prod_{t=1}^{T}\Pr(o_t|q_t,\lambda)$, where $O$ is observation and $Q$ is state sequence. and I guess $\lambda$ is the initial distribution of HMM. I don't understand why he put this equality. ref:digital.cs.usu.edu/~cyan/CS7960/hmm-tutorial.pdf, Page 3 $\endgroup$ – Lvergergsk Oct 26 '15 at 7:17
  • $\begingroup$ @Lvergergsk. I think by $Pr(x)$ they mean $Pr(X=x)$. When you have only one variable $X$ such a collocation could be reasonable. $\endgroup$ – Nikita Evseev Oct 26 '15 at 7:26

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