![]() We asked whether the brain avoids combining measurements when they differ from one another: that is, whether the system is robust to outliers. However, the system should only combine measurements when they come from the same source. Previous work has shown that the perceptual system combines measurements using a statistically optimal weighted average. Higher degrees of freedom, less uncertainty, t-distribution will look more and more like a normal distribution.Depth perception involves combining multiple, possibly conflicting, sensory measurements to estimate the 3D structure of the viewed scene. lower degrees of freedom, more uncertainty, fatter tails. So even if you standardize the variable, the distribution will depend on the degrees of freedom of the distribution. The t distribution "ratio" is larger than the one for normal distribution. In other words, once you standardized a variable, (so that it has a mean zero and standard deviation 1), if that variable follows a normal distribution, it will always look the same. However, a normal distribution has a very nice "ratio" of dispersion around the tails, and concentration around the means. In a normal distribution, the larger the variance, the farther the tails will spread. However, the larger the sample is, the more accurate the Standard deviation estimate will be, thus t-distribution will converge with the normal distribution. Since we do not know the standard errors, there is more uncertainty of the t-statistic. Why is t-distribution fatter? because there is more uncertainty. ![]() Your "title" question seems more clear to me than your detailed question. In my original idea of being on tail I would consider the area $ and $[\mu 3\sigma, \inf)$ and if some distribution has "fatter tail" would be to have large area under the curve in that tail area. So I wonder why Wikipedia has this definition of being "fat". In case of Exponentiation distribution 3rd and 4th moments are: 2 and 6, and in case of Normal distribution those moments are zero. It is pretty unclear why both the normal distribution or an exponential distribution are considered in here because their third and forth moment formula differs. So my hypothesis H looks like is wrong at the very start, because you cannot alter the Normal distribution "fat tail" state with the standard deviation parameter. In general, and I understood this for the first time thanks to " a fat-tailed distribution" is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution according to Wikipedia It looks to me the answer is yes, for the previous question but maybe I am missing something important. Q: Is this comparison based on the normal distribution used to produce the t-distribution? H: In general you can control the tails of normal distribution with standard deviation, the bigger it is the fatter the tails so I can make the tails as fat as I like. I need to clarify the Hypothesis H and Question Q: and the population's standard deviation is unknown.when estimating the mean of a normally-distributed population.
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