Suppose X is a random variable with the pdf fx which is symm

Suppose X is a random variable with the pdf f(x) which is symmetric about 0, i.e. f(-x) = f(x). Show that F(-x) = 1-F(x), for all x in the support of X

Solution

we have the pdf of f(x) that is symmetric about 0 so that means f(-x) = f(x)

if we derivate f(x) we will have F(X)

if we derivate f(-x) we will have F(-x)

Thus, X and X have the same distributions. Define Xn as

Xn = X if n is odd

Xn = -X if n is even

early, FXn (x) = FX(x) and Xn D X. However, Xn P X

so that means that F(-x) will be the same as say 1 - F(x)