What is the distribution of absolute values of a random normal variable?

Answer 1

It is the so-called "half-normal distribution."

Specifically, let X be a standard normal variate with cumulative distribution function F(z). Then its cumulative distribution function G(z) is given by

Prob(|X| < z)

= Prob(-z < X < z)

= Prob(X < z) - Prob(X < -z)

= F(z) - F(-z).

Its probability distribution function g(z), z >= 0, therefore equals

g(z) = Derivative of (F(z) - F(-z))

= f(z) + f(-z) {by the Chain Rule}

= 2f(z)

because of the symmetry of f with respect to zero. In other words, the probability distribution function is zero for negative values (they cannot be absolute values of anything) and otherwise is exactly twice the distribution of the standard normal.