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چکیده
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Let $X_1,\ldots,X_n$ be a set of $n$ continuous and non-negative random variables, with log-concave joint density function $f$, faced by a person who seeks for an optimal deductible coverage for this $n$ risks. Let ${\bf d}=(d_1 , \ldots d_n)$ and ${\bf d}^*=(d^*_1 , \ldots d^*_n)$ be two vectors of deductibles such that ${\bf d}^*$ is majorized by ${\bf d}$. It is shown that $\sum_{i=1}^{n} (X_i\wedge d_{i}^*)$ is larger than $\sum_{i=1}^{n} (X_i\wedge d_{i})$ in stochastic dominance, provided $f$ is exchangeable. As a result, the vector $(\sum_{i=1}^{n}d_i, 0,\ldots,0)$ is an optimal allocation that maximizes the expected utility of the policyholder's wealth. It is proven that the same result remains to hold in some situations if we drop the assumption that $f$ is log-concave.
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