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| -rw-r--r-- | ps1/main.tex | 4 |
1 files changed, 2 insertions, 2 deletions
diff --git a/ps1/main.tex b/ps1/main.tex index 35ad24b..ac1a2b9 100644 --- a/ps1/main.tex +++ b/ps1/main.tex @@ -285,7 +285,7 @@ a simpler form. \end{cases} \end{displaymath} then our optimization problem becomes a fractional Knapsack problem for - which the optimal solution is easy to compute: order agents by + which the optimal solution is easy to compute: order agents by decreasing order of expected values and allocate to them in this order until the budget is exhausted. \end{enumerate} @@ -339,7 +339,7 @@ Since $\sum_i v_i$ is a symmetric random variable centered at $\frac{n}{2}$. Now we have to compute $\E\left[I_n\sum_{i=1}^n v_i\right]$. It is hard to obtain a closed-form formula, but we can obtain an asymptotic approximation -when $n$ goes to infinity by observing that the sum of iid variables behaves +when $n$ goes to infinity by observing that the sum of i.i.d. variables behaves like a normal distribution in the limit. This can be made formal by applying the central limit theorem. Let us define: \begin{equation}\label{eq:xn} |