Compute and interpret quantile loss vs RMSE

Define the quantile (pinball) loss L_q(y, ŷ) for quantile q and explain how it evaluates probabilistic forecasts differently from RMSE/MAE. Given observations y = [100, 120, 90] and predicted 90th-percentile forecasts ŷ_q = [110, 115, 95] at q = 0.9, compute the average pinball loss. Then compute the RMSE for median forecasts ŷ_0.5 = [102, 118, 93]. Under asymmetric underprediction costs, decide which model you would prefer and justify. Finally, discuss methods to assess calibration and empirical coverage of 80% and 90% intervals.