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Prove that the characteristic function of any random variable must

Prove that the characteristic function of any random variable must satisfy the following properties.

(a) ϕ*X (ω) = f X (– ω).

(b) ϕX( 0) = 1.

(c) For real ω, |f X (ω) = 1.

(d) If the PDF is symmetric about the origin (i. e, an even function), then ϕX(ω) is real.

(e) ϕX( .) cannot be purely imaginary.

(a) ϕ*X (ω) = f X (– ω).

(b) ϕX( 0) = 1.

(c) For real ω, |f X (ω) = 1.

(d) If the PDF is symmetric about the origin (i. e, an even function), then ϕX(ω) is real.

(e) ϕX( .) cannot be purely imaginary.

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