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
- define an experiment/random phenomenon (e.g measuring the temperature in my room);
- determine the Sample Space (e.g. Set Power(N)={N,{1},{2},..{1,2},{1,3}...};
- assignment of probabilities to the each events in the Sample Space (e.g. in Set Power(N))
- 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.
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
$$P(X=i) \geq 0 ~\forall i \in \{-10, \ldots, 40\}.$$