this is maybe a silly question but I do not have any idea how to solve it. Given an unknown distribution with mean $E(x) = 0$ and variance $Var(x) = \sigma^2$, can I derive from these two parameters that my distribution is a normal distribution?
Thanks for your answers!
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$\begingroup$Any distribution at all can be translated to have a mean $E(x)=0$ without changing its shape. Furthermore, any distribution with a non-zero variance can be scaled to have whatever variance you choose.
Therefore simply knowing the mean and variance of a distribution is not nearly sufficient to identify the distribution. It could be normal; it could be uniform; it could be..... pick a shape.
$\endgroup$ 3 $\begingroup$You can build a histogram over your data and compare it to the expected histogram (of a Gaussian distribution) from a given $\mu$ and $\sigma^2$. One measure you can try is for example the famous Kullback-Leibler Divergence:
$$KL(p,q) = \int_{-\infty}^{\infty}p(x)\log\left(\frac{p(x)}{q(x)}\right)dx$$
Which for your histograms $h_p,h_q$ can be estimated as a Riemann-sum.
$$KL(p,q) \approx \sum_{k}h_p[k]\log\left(\frac{h_p[k]}{h_q[k]}\right)dx$$
$\endgroup$ $\begingroup$Well for one thing, this definitely isn't possible (as said by a few answers already). If it were, then datasets would only have 2 free variables (mean and variance)! Clearly, there are more free variables than two, so there are many more things at play here. Also, if one could immediately determine the distribution of a dataset so easily, stats and data science would be a lot easier.
In addition to some of the things said by other answers, you can also use the Probability Plot. It allows you to test your dataset's histogram against a distribution. It's a nice visualization and lets you know if your dataset fits the model you've predicted.
Lastly, if you determine the most likely type of distribution (say, exponential, geometric, normal etc.), then you can usually determine its parameters if you know the mean and variance of your data by using the maximum likelihood estimates (which are available online for most standard distribution families)
Hope that helps!
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