It is proved that the characteristic function of continuous random variable x is a real function if and only if its density function f (x) is symmetric, that is, f (x) = f (- x)

It is proved that the characteristic function of continuous random variable x is a real function if and only if its density function f (x) is symmetric, that is, f (x) = f (- x)

That is ∫ (0 → + ∞) sin (UX) [f (x) - f (- x)] DX = 0, so f (x) - f (- x) = 0, that is, f (x) = f (- x), that is, f (x) is even function. Therefore, the necessary and sufficient condition for the characteristic function of continuous random variable x to be a real function is that its density function f (x) is even function