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For probabilities $p_i=\frac{i}{10}$ where $i=1, \dots, 10$, the respective quantiles are $\tau_i$. How can I calculate an approximate expected value?

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Using my answer at Expected value as a function of quantiles?, a general expression for the expectation in terms of the quantile function is $$ \mu=\int_0^1 Q(p)\; dp $$ in the continuous case, and that answer extends to the general case.

Looking at the approximating sums defining the integral, you can read this as the mean is the mean of the quantiles, which gives an approximation for your case as $$ \frac{\sum_1^{10} \tau_i}{10} $$

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    $\begingroup$ Isn't that slightly biased upwards due there being no $Q(0)$ and if yes, do you see a way of correcting that? $\endgroup$ Dec 7, 2022 at 14:16
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    $\begingroup$ @LukasLohse: Yes, but this comes from the biased definition of $p_i$ by the OP $\endgroup$ Dec 7, 2022 at 14:19
  • $\begingroup$ @LukasLohse Use the i's from 1 to 9? $\endgroup$ Dec 7, 2022 at 14:32
  • $\begingroup$ @LukasLohse: I guess this must be adapted to the specific def of quantile used ... it doesn't seem right to leave out the top quantile completely $\endgroup$ Dec 7, 2022 at 14:37
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    $\begingroup$ @user529295: What do you mean by random quantiles? $\endgroup$ Dec 7, 2022 at 17:46

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