Stochastic models
I made progress thinking about probability distributions today. I woke up thinking about distributions and then I started thinking
about the fundamental axioms and definitions of probability and used them to derive the probabilty mass function for a binomial distribution. The proof helped me to clarify some terminology. A random variable variable is a member of set of experiments. An experiment is series of events.
eg. E = {A,B} where A and B are events.
An experiment is a series of events like this AAABB. this means A happened 3 times followed by B happening 2 times. A probability of a experiment is given by the amount of times of the occurance of such an experiment divided by the total size of the sample space for that experiments. An experiment with 3 trials with n(E)=2 has a sample space size of 2^3=8. The number of times of of saying obtaining 2A's in such experiment is 3C2*Pr(A)^2*(P(B))^1. This is exactly the form of probability mass function for binomial distribution.
It therefore follows by studying such a function one can tell by sampling and other methods(i think chi test) whether a series of randoms values(aka random variables) has the same function or not. This is an amazing find when I come to think of it. This is the heart of statistical modeling. You start with data, and you model it using a probability distribution(i.e probability mass function), and you use function to predict how it will behave in the future. If you model is incorrect there will be discrepancy between the data and model, and hence you back to the drawing table of coming up with another probability distribution to model the data.
about the fundamental axioms and definitions of probability and used them to derive the probabilty mass function for a binomial distribution. The proof helped me to clarify some terminology. A random variable variable is a member of set of experiments. An experiment is series of events.
eg. E = {A,B} where A and B are events.
An experiment is a series of events like this AAABB. this means A happened 3 times followed by B happening 2 times. A probability of a experiment is given by the amount of times of the occurance of such an experiment divided by the total size of the sample space for that experiments. An experiment with 3 trials with n(E)=2 has a sample space size of 2^3=8. The number of times of of saying obtaining 2A's in such experiment is 3C2*Pr(A)^2*(P(B))^1. This is exactly the form of probability mass function for binomial distribution.
It therefore follows by studying such a function one can tell by sampling and other methods(i think chi test) whether a series of randoms values(aka random variables) has the same function or not. This is an amazing find when I come to think of it. This is the heart of statistical modeling. You start with data, and you model it using a probability distribution(i.e probability mass function), and you use function to predict how it will behave in the future. If you model is incorrect there will be discrepancy between the data and model, and hence you back to the drawing table of coming up with another probability distribution to model the data.
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