Chapter 1: Sub-Gaussian Random Variables
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چکیده
where μ = IE(X) ∈ IR and σ = var(X) > 0 are the mean and variance of X . We write X ∼ N (μ, σ). Note that X = σZ + μ for Z ∼ N (0, 1) (called standard Gaussian) and where the equality holds in distribution. Clearly, this distribution has unbounded support but it is well known that it has almost bounded support in the following sense: IP(|X −μ| ≤ 3σ) ≃ 0.997. This is due to the fast decay of the tails of p as |x| → ∞ (see Figure 1.1). This decay can be quantified using the following proposition (Mills inequality).
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