1 Is the real valued function fx R arrow R defined by fxx a

1. Is the real valued function fx: R arrow R defined by fx(x)= a pdf for some continuons random variable X? What about gx (x) defined below? gx(x)=

Solution

1.

A pdf must have an integral of 1 for all real x.

Thus,

Integral [fX(x) dx] (-oo, +oo) = Integral [2x^-2 dx], (1,2)

= -2x^-1 | (1,2)

= [-2/2] - [-2/1]

= -1 + 2

= 1 [IT IS A PDF!]

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For gX(x), note that

|x| = x for x > 0,
-x, x < 0

Thus,

Integral [fX(x) dx] (-oo, +oo) = Integral [(1 - x)dx], (-1, 0) + Integral [(1 + x)dx], (0, 1)

=[x - x^2/2] (-1, 0) + [x + x^2/2] (0, 1)

= 3 [IT\'S NOT A PDF!]