What is the meaning of the difference between the quantile of prob=0.05 and mean for a sample form a specific distribution?
In other words, I would like to understand the relationship between quantile and mean? Thanks a lot.
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I think first understanding what the mean is and what a quantile is would be helpful. The 0.05 quantile is the value for which for a given distribution only 5% of the values are expected to be lower, for example in the standard normal distribution this quantile is roughly -1.65. Now to understand a relationship between those two you have to look at the graphs of distributions and think if those two values are close this means that a lot of mass is "near" 5% quantile roughly speaking and for unimodal distributions, this means that our distribution is right-skewed. Find a way to plot all those distributions that you have and this will help you get a better understanding of the topic.
P.S VaR i.e Value-at-Risk and Var i.e variance can be easily confused so try to distinguish them more clearly.