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Let $X_1$, $X_2$, ..., $X_n$ be independent normal variables with zero mean and decreasing variances: $X_i=N(0,v_i)$ where $v_1>v_2>...>v_n\ge 0$. Assume $n>2$. Let $p_i=Prob(X_i = max(X_1,X_2,...,X_n))$.

Prove the following
Conjecture: $p_1>p_2>...>p_n$.

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    I submitted & deleted this erroneous answer, but am editing out the erroneous part for what its worth 2. $p_i = \int \frac {\phi_i(x)} {\Phi_i(x)} \prod \limits_{j=1}^n \Phi_j(x))$ which is of the form $ \int \frac {\phi_i(x)} {\Phi_i(x)} h(x), h \uparrow$. Fix all parameters that occur in $h$ and look at this as $p(t) = \int \frac {t\phi(t x)} {\Phi(t x)} h(x)$. Then $p_i = p(\frac 1 {\sigma_i})$. $\frac {t\phi(t x)} {\Phi(t x)} = \frac d {dx} $ so $p(t) = -\int log \Phi(t x) h^{\prime} $, $p^{\prime}(t) = -\int (\frac d {dt} {log \Phi(t x)} ) h^{\prime}(x)$2012-12-04

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This is not rigorous, but consider asking for $P(X_i > t)$ for some fixed $t$. If all the $X$'s are normal with mean 0, then we can convert all of them to standard form by dividing by the standard deviations, which are also decreasing. So we can now ask for $P(Z_i > \frac{t}{\sigma_i})$. But $Z_i$ is increasing in $i$ as the numerator is fixed and the denominator is decreasing in $i$. In other words, we have the sequence $Z_1 < Z_2 < Z_3 \ldots$. Therefore, the probability of getting $X_i > t$ is $1 - \Phi(Z_i)$ and this decreases as $Z$ increases. Therefore, it can be shown for some large $t$ that $X_1$ has the largest probability of meeting or exceeding it, so probabilistically it should have the max, and the probability of exceeding a given number is strictly decreasing in the $X_i$'s as long as the variance is strictly increasing.