# About random variable and probability notation

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.

• 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 $Pr(X=x) = Pr (A)$. – kmitov Oct 26 '15 at 6:46

## 1 Answer

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.

• Thanks for your and other replies. Lots of papers just write $\Pr(X)$ and I'm really confused with that. – Lvergergsk Oct 26 '15 at 7:01
• @Lvergergsk Well, it could depends on context. Would you show an example? – Nikita Evseev Oct 26 '15 at 7:05
• 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 – Lvergergsk Oct 26 '15 at 7:17
• @Lvergergsk. I think by $Pr(x)$ they mean $Pr(X=x)$. When you have only one variable $X$ such a collocation could be reasonable. – Nikita Evseev Oct 26 '15 at 7:26