I have been studying statistics lately, and after having my first brush with normal distributions, I was curious to know more about it. I have researched about the history of normal distributions and found the following information:
When gauss published his monograph, Theoria motus corporum coelestium in sectionibus conicis solem ambientium, he discussed among other things the normal distribution (interestingly, the book was on how celestial objects have orbits that are conic sections). According to Wikipedia, he did the following to arrive at the distribution:
There is an unknown quantity $V$ . $M, M',M",\dots$ are the measurements of $V$. $\varphi$ is the function that governs the law of probability. The aim was to find $\varphi$. It says on Wikipedia that the most likely estimator of $V$ would be the one that maximizes:
$$\varphi(M-V)\varphi(M'-V)\varphi(M"-V)\dots\;\;\;\;\;\;\;\;\;\;-(1)$$ Gauss guessed that the answer to the $V$ maximizing the above equation must be the mean of the measured values. Then gauss demonstrated that the only function $\varphi$ that gives the mean as the answer was:
$$\varphi\Delta=\frac{h}{\sqrt{\pi}}e^{-h^2{\Delta}^2}$$ where,
$\Delta$ is the measurement of errors.
$h$ is the measure of precision of the observation
A number of questions pop up in my head.
- Why should the most likely estimate be the one that maximizes equation 1?
- How did Gauss' derive $\varphi$ using the assumed solution to be the mean?
- What does $\Delta$ actually mean?
MY ANSWERS/ATTEMPTS
I actually feel that it should not be the maximum. It should rather be the minimum as when the expression is minimized it means that $V$ is closer to each of the values $M,M',M"\dots$ which are the actual measurements. Therefore such a value should be the most likely estimate. From my knowledge of statistical inference, in the procedure of maximum likelihood estimator, we maximize the probability density function of the distribution. The value of random variable at the maxima is the estimate.
I do not know the answer to this one. I tried looking up Gauss' original work but it would take me too off topic as Gauss' has treated these ideas with respect to orbits of celestial objects being conic sections. I do not wish to look into the applications right now (But I do understand that reading the book right from the start will be wonderful learning experience).
I once again do not fully understand what $\Delta$ means here. According to me, $\varphi$ should have been a function of $V$ and not $\Delta$.
It would be really helpful if someone could answer my questions and also provide an explanation of what actually is being done here as I am getting confused with so many ideas in my head!!