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Is there a deep difference between a Normal and a Gaussian distribution, I've seen many papers using them without distinction, and I usually also refer to them as the same thing. However, my PI recently told me that a normal is the specific case of the Gaussian with mean=0 and std=1, which I also heard some time ago in another outlet, what is ...
Due to the Central Limit Theorem, we may assume that there are lots of underlying facts affecting the process and the sum of these individual errors will tend to behave like in a zero mean normal distribution. In practice, it seems to be so. I'm interested in the second part actually.
$\begingroup$ Only the errors follow a normal distribution (which implies the conditional probability of Y given X is normal too). This is probably traditional because of reasons relating to the central limit theorem. But you can replace normal with any symmetric probability distribution and get the same estimates of coefficients via least squares.
Yes, you can, for precisely the reason you give: even if the underlying population is not normally distributed, the mean (or more precisely the difference between the means) is asymptotically normal. (There are some conditions on the underlying populations that are usually satisfied in the real world, and certainly for underlying uniform ...
What is meant by the statement that the kurtosis of a normal distribution is 3. Does it mean that on the horizontal line, the value of 3 corresponds to the peak probability, i.e. 3 is the mode of the system? When I look at a normal curve, it seems the peak occurs at the center, a.k.a at 0. So why is the kurtosis not 0 and instead 3?
Here are two normal and gamma distribution relationships in greater detail (among an unknown number of others, like via chi-squared and beta). First A more direct relationship between the gamma distribution (GD) and the normal distribution (ND) with mean zero follows. Simply put, the GD becomes normal in shape as its shape parameter is allowed ...
Please be aware that "if inferences are drawn about the relationship (e.g. we set a null hypothesis that r = 0; [no correlation]), then the Pearson's correlation coefficient assumes that the joint distribution of X and Y is ‘bivariate normal’.
That slightly differs from a normal distribution and the shapiro.test also rejects the null hypothesis that the residuals are from a normal distribution: > shapiro.test(residuals(lmresult)) W = 0.9171, p-value = 3.618e-06 The residuals vs fitted values look like: What can I do if my residuals are not normally distributed?
Then they would be independent, and they would have the same distribution, but that distribution would not be normal. On the other hand, a set of data can be normally distributed, but with subsets that follow different normal distributions (e.g., different means or different variances). In addition, you could have two subsets that are dependent ...