validity of a PDFfunction .

Question

For z∈R , define the following two functions:

$f_1(z)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{\max (1,z^2)}{2}\right)$

$f_2(z)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{\min (1,z^2)}{2}\right).$

i want to know if any of the functions is a valid PDF or can be if multiplied by a non negative constant c .

the way i see it is that the first one is a valid PDF if multiplied by constant.

but i'm still confused about the second one ?

Answer 1

$$\int_1^\infty\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{\min (1,z^2)}{2}\right) \, dz = \int_1^\infty\frac{1}{\sqrt{2\pi }}\exp \left(-\frac1{2}\right) \, dz=\infty$$

Hence, it can't be multiplied by a positive constant and form a pdf.