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I have a gaussian random variable N(0,sigma)

I know how to calculate probabilities like P(N>value) or P(N < value) thanks to Q-function.

I'd like to know how to calculate probabilities like P(sqrt(N)>value) or P(sqrt(N) < value).

And also probabilities like P(N^2>value) or P(N^2 < value)

Thanks really.

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1 Answer 1

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That requires you to have the distribution of the transformed random variable. If $N$ is normal with mean zero and a certain variance, you find the distribution of its square, e.g., here: Square of normal distribution with specific variance

As for the square root, that transformation is problematic, as a normal random variable has support on the entire real line, and that would amount to the possibility of taking the square root of a negative number.

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