Prove that the characteristic function of any random variable must satisfy the following properties (a) X () f X ( ) (b) X( 0) 1 (c) For real , f X () 1 (d) If the PDF is symmetric about the origin (i e, an even function), then X() is real (e) X( ) cannot be purely imaginary ...

a Since the pdf is a real function its complex conjuate will b ...

Prove that the characteristic function of any random variable must satisfy the following properties. (a) *X ()

Question:

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.