Please answer the following question with explanation. thank you

Problem 4. Let X : 0 -> R be a continuous random variable on a probability space (0, F, P) with cumulative distribution function F : R -> [0, 1] and characteristic function ? : R -> C. (1) (5p) Show that the characteristic function $ : R -> C of -X satisfies 4(() = 4(() for all SER. (2) (7p) Show that one has (() ( R for all ( ( R if and only if F(c) = 1 - F(-c) for all ce R. (3) (8p) Let Y : 0 - R be a random variable that is independent of X but has the same distribution as X. Set Z := X -Y, and let G : R -> [0, 1] be the cumulative distribution function of Z. Show that G(c) = 1 -G(-c) for all ce R