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1. ### Right and left quantiles

Hi all, I am struggling to prove left and right quantile relationship: Right quantile (0.95) (x) = -left quantile (1-0.95) (-x) I tried to prove with probability manipulations but struggle to get correct result. Could you please help? Thank you
2. ### Kurtosis with stochastic volatility

I am struggling to prove that for a normally distributed loss RV introducing stochastic volatility (\sigma_1 with probability 0.5 and \sigma_2 with probability 0.5) would make kurtosis bigger than 3 (fat tails). Can someone help? Thank you
3. ### Kurtosis empirical evidence

Hi All, If we are given Loss RV as : How to prove that kurtosis is 3? Additionally if Volatility is stochastic: How to prove that the distribution is fat tailed (kurtosis is greater than 3) assuming and are independent? Thank you
4. ### Logistic Distribution

Hi All, If .the logistic distribution is defined as it corresponding quantile function: How can I show that q is strictly increasing and compute logistic distribution function and it's density function? How to compute VaR and ES of r.v. with logistic distribution? Thank you for your help