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Keep one distribution and confidence
The original losses −10/0/10/50 carry probabilities 30/40/25/5 percent. The lower 90% loss quantile is 10. The boundary loss itself has positive mass, so different subset rules select different probabilities.
Name each denominator
| Definition | Selected probability | Mean loss |
|---|---|---|
| Exactly worst 10% | 10% | 30 |
| Conditional on L > 10 | 5% | 50 |
| Conditional on L ≥ 10 | 30% | 16.666667 approximately |
Calculate the inclusive average
The inclusive subset contains loss 10 with 25% and loss 50 with 5%. Its weighted loss sum is 0.25×10+0.05×50=5. Divide by 0.30 to get 16.666667 approximately. The strict subset contains only 50.
Keep definitions explicit
For a continuous distribution without threshold mass, familiar conditional-tail descriptions can agree with the exact selected-tail measure. At atoms, state the convention before calling those averages identical. A zero-probability conditioning event also needs separate treatment.