Suppose a vector X of length n, where each component of X is normally distributed with mean 0 and variance 1, and independent of the other components. I want to know the probability that at least one of X_1>2, X_2>2.5, X_3>1.9, etc. happens (inclusive), i.e. the probability that the vector's individual values are all greater than a vector with some arbitrary values. In the above case, I'd be looking for the probability that the individual components of X are greater than [2, 2.5, 1.9, ... ]. To me this looks like a CDF, but a CDF of n independently distributed random variables. Does this have a closed form? If not, is there a manageable way to approximate it? Would it have a sigmoid shape (i.e. concave where all components are positive in n-space)?(adsbygoogle = window.adsbygoogle || []).push({});

Actually, I think the vector notation confuses things. So basically, I'm just asking about the "CDF" of a bunch of independent random variables that all are distributed standard normal.

Thanks.

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# CDF of a normal vector is there a closed form?

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