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I have to define a function using mathematical notation.

A function is translating one molecule ID to a vector(array,one dimensional matrix) of states that it can be in.

I basically need to say the following: For each of the molecule(in the set of molecules IDs, lets call it M) there exists a function that is translating its ID to a set of the probabilities(Real numbers from 0 to 1) of its states(states are defined in the set S).

Thank you in advance.

M.

user2703038
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1 Answers1

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If I'm interpreting the situation correctly, let me give you my take on the an appropriate mathematical structure to analyze the situation:

There is some experiment (whether just happenstance of the universe, or controlled in a lab) where certain molecules are produced where each molecule you are considering may have some different arrangements or realizations. Let $\Omega$ be the collection of all possible outcomes of this experiment (that is, $\Omega$ will contain all the different realizations of the molecule(s) you are considering, so you can interpret an element $\omega$ in $\Omega$ as one particular realization of the molecule). We will call $\Omega$ the sample space as it contains all the samples you are interested in. Depending on the different realizations of each $\omega \in \Omega$, one question you can ask is "what is the particular state that $\omega$ is in?" Let $S$ be the collection of all possible states which answer this question. We will call $S$ the state space. Let $X$ be a function from $\Omega$ to $S$ (written $X : \Omega \to S$) such that for each $\omega \in \Omega$, $X(\omega)$ outputs the state that $\omega$ is in. In the language of probability, $X$ is a random variable. I will assume that $S$ has only finitely many states (or at most countably many), in which case, associated to $X$ is a probability mass function $p_X$ which takes in elements from the state space $S$ and outputs values in $[0,1]$ (again, written as $p_X : S \to [0,1]$) where if $s \in S$ is one of the possible states, then $p_X(s)$ gives the probability that a randomly selected molecule is in state $s$. Again using the common symbols of probability, we write $p_X(s) = P(X=s)$, which asks for the probability that if you randomly select $\omega \in \Omega$, then $X(\omega)=s$. Now, what you're asking for sounds a lot like this probability mass function $p_X$.

Tom
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  • Thank you Tom, your explanation is clear and straightforward. – user2703038 Nov 17 '13 at 20:52
  • Hi Tom, one more question. The pmf function requires that the sum of it's values add to 1. In the context of applying the probability vector to a model that poses a problem. The probability vector is there to impose an constraint to the possible states. Hard limit will say that the states that have the probability of 0 are not allowed and those with the probability of 1 are allowed. The soft limit (for example 0.5) imposes a preferability of one state over another. – user2703038 Nov 17 '13 at 21:25
  • The probability vector defines the probability of the states being regarded as valid or invalid (hard limit) or to define a preferred state using soft limit. – user2703038 Nov 17 '13 at 23:52
  • @user2703038 I apologize; I'm afraid that I don't understand what it is you're looking for then. Certainly the elements of a pmf sum to $1$ since this is equivalent to the statement, "The probability that I randomly select some $\omega \in \Omega$ and it belongs to any of the possible states is $1$". I'm not sure what is meant by a hard limit vs. soft limit in your model, nor what sort of constraints are made by a probability vector. Good luck, and if anything dawns on me, I'll let you know. – Tom Nov 18 '13 at 12:40