How to find the maximum of Gaussian random variables?
Giving (upper) bounds on the maximum of i.i.d Gaussians is easier than precisely characterizing its moments. Here is one way to go about this (another would be to combine a tail bound on Gaussian RVs with a union bound). Let Xi for i = 1, …, n be i.i.d N(0, σ2).
Which is the maximum of IID normal variates?
A suitably standardized maximum of iid Normal variates converges to a reversed Gumbel distribution. When the Xi are iid with common distribution function F, the distribution of the maximum X ( n) is Fn(x) = Pr (X ( n) ≤ x) = Pr (X1 ≤ x) Pr (X2 ≤ x)⋯ Pr (Xn ≤ x) = Fn(x).
When does fn have an extreme value distribution?
When F is a Normal distribution, the particular limiting extreme value distribution is a reversed Gumbel, up to location and scale. It is tempting to emulate the Central Limit Theorem by standardizing Fn to have unit mean and unit variance.
Which is the maximum of the Gumbel distribution?
The Maximum of X1, …, Xn. ∼ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We need to find/choose sequences of constants such that: F(anx + bn)n →n → ∞G(x) = e − exp ( − x)
How to calculate the mean and variance of a random variable?
If x ↦ f(x) is the density function of a random variable X with expected value 0 and variance 1, then x ↦ 1 σf(x − μ σ) is the density function of μ + σX, and thus of a random variable with expected value μ and variance σ2. That can be shown by thinking about the substitution u = x − μ σ and du = dx σ.
How to derive the derivative of a Gaussian random?
The derivative of ∫ ∞ − ∞ exp { − u2 }du is in the form of a Gaussian integral. I’m not going to pretend that I can easily derive it (multivariable calc???) but it is widely known that the value of this integral is √π.