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A random variable X is called sub-gaussian if
[tex] P( |X| > t) \leq 2e^{-K t^2} [/tex]
for some constant K. Equivalently, the probability density p(x) is sub-gaussian if
[tex] \int_{-t}^{t} p(x) dx \geq 1 - 2 e^{-Kt^2}. [/tex]
The challenge: Prove that the standard normal distribution (with mean 0 and variance 1) is sub-gaussian.
[tex] P( |X| > t) \leq 2e^{-K t^2} [/tex]
for some constant K. Equivalently, the probability density p(x) is sub-gaussian if
[tex] \int_{-t}^{t} p(x) dx \geq 1 - 2 e^{-Kt^2}. [/tex]
The challenge: Prove that the standard normal distribution (with mean 0 and variance 1) is sub-gaussian.