A random variable is called symmetric if, for any interval a b, P(a X b) = P(b X

Question

A random variable is called symmetric if, for any interval a b,

P(a X b) = P(b X a).

(a) Suppose P(X = x) = 0 for all x R. Show that X is symmetric if and only if FX(a) + FX(a) = 1 for all a R. [Hint: to show that symmetry implies this condition, let b + in the definition of symmetry.]

(b) If X is symmetric and has a continuous density fX, show that the random variable Y = X2 has density given by fY (x) = fX( x) x , x > 0 and fY (x) = 0 for x < 0. [Hint: compute FY (x) in terms of FX, simplify using part (a), and differentiate.]

(c) Suppose X is a standard normal random variable: fX(x) = 1 2 e x 2/2 . What is the density of X2 ?

Explanation / Answer

P(a X b) = Fx(b)-Fx(a) & P(b X a)=Fx(-a)-Fx(-b)=Fx(b)-Fx(a). Then consider P(a X a)= Fx(a)-Fx(a)= Fx(a)+Fx(-a)=1.

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