Sensor and Random Variable

Please, let us imagine that I have got a sensor D that measures the temperature in my room. My issue and questions are: some papers claims to model it as a Random Variable (RV) X. What does it mean? My understanding is that to model it as a RV we need to consider a Probability Space as in the following

1. define an experiment/random phenomenon (e.g measuring the temperature in my room);
2. determine the Sample Space (e.g. Set Power(N)={N,{1},{2},..{1,2},{1,3}...};
3. assignment of probabilities to the each events in the Sample Space (e.g. in Set Power(N))
4. then, I define a Random Variable X as a function from the Sample Space to the Real field R, e.g. , ,etc

I am confused. Please could you provide any clarification to the subject? I would really appreciate. Many Thanks.

• In order for $X$ to be a random variable, the sample space, the sigma algebra, and the probability measure have to exist, but that doesn't mean you know what they are, or indeed that you have any way of knowing what they are. In this case, you may know what the sample space and sigma-algebra are, but you may not know what the probability measure is. (And yeah, the function representing the random variable itself. I forgot about that.) – Brian Tung Mar 8 '19 at 19:58
• Let $\Omega$ be all the possible "states"of your room at any given time. Then, given any state $\omega$, define $X(\omega)$ to be the temperature associated to that state. – rubikscube09 Mar 8 '19 at 19:58
• Thank you for your answers. What is state? – Carlo Allocca Mar 8 '19 at 20:09

I will try to give you an intuition about random variables.

I think that the easiest way to talk about RVs is the dice. A RV representing a dice is a variable $$X$$ which assumes values in the set $$\{1, 2, \ldots, 6\}$$. For each outcome of $$X$$, you can associate a probability. Specifically:

$$P(X = 1) = \frac{1}{6}, P(X=2) = \frac{1}{6}, \ldots ~\text{and so on.}$$

If $$X$$ is the room temperature measured by a sensor, then the set of $$X$$ is composed by all the possible temperature values that your sensor can measure. For example, using Celsius degree, and supposing that your sensor is digital and can provide only integer temperature between $$-10$$ and $$+40$$, we can say that:

$$X \in \{-10, -9, \ldots, 0, 1, \ldots, 39, 40\}.$$

While for the dice, the "state" is the "face of the dice", for the temperature the "state" is given by the measurement provided by the sensor, which is random. The set $$\{-10, \ldots, 40\}$$ is the "sample space".

Now, for each element of this set, a probability must be specified. If $$P(X = a)$$ is the probability that the sensor returns $$i$$, we must satisfy the followings:

$$\sum_{i=-10}^{40} P(X = i) = 1,$$

and

$$P(X=i) \geq 0 ~\forall i \in \{-10, \ldots, 40\}.$$

• Thanks. But if we don t define a probability function as for the case of the dice, can we call X RV? – Carlo Allocca Mar 8 '19 at 20:21
• An RV need the sample set and the associated probabilities. – the_candyman Mar 8 '19 at 20:36
• So, where I get the probability from ? Or I cannnot define X as a RV? – Carlo Allocca Mar 8 '19 at 20:40
• @CarloAllocca If something is random, then a probability distribution is present. Maybe the problem you are facing ask you to find a "reasonable" probability distribution for the problem. For example, if we are in summer, I guess that $P(X = 0) = 0$, while $P(X = 25)$ is quite high. – the_candyman Mar 8 '19 at 20:47
• @CarloAllocca You asked some clarification on RV. If you want to solve with a specific problem, please be more precise. You can ask another question, but please be sure that you add all your efforts to solve that problem. – the_candyman Mar 8 '19 at 20:51