## Calculus with Applications (10th Edition)

$$f\left( x \right){\text{ }}is{\text{ }}not{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function$$
\eqalign{ & f\left( x \right) = 4{x^3};\,\,\,\,\,\,\,\left[ {0,3} \right] \cr & {\text{The function f is a probability density function of a random variable X in the interval }} \cr & \left[ {a,b} \right]{\text{ if}} \cr & 1{\text{ condition}}:f\left( x \right) \geqslant 0{\text{ for all }}x{\text{ in the interval }}\left[ {a,b} \right].{\text{ then}} \cr & 4{x^3} \geqslant 0 \cr & {x^3} \geqslant 0 \cr & x \geqslant 0 \cr & \left[ {0,3} \right]{\text{ is on the interval }}\left[ {0,\infty } \right).{\text{ then }}f\left( x \right) \geqslant 0{\text{ for the interval }}\left[ {0,3} \right] \cr & \cr & 2{\text{ condition}}:\int_a^b {f\left( x \right)} dx = 1.{\text{ then}} \cr & \int_0^3 {\left( {4{x^3}} \right)} dx \cr & {\text{integrating}} \cr & = \left( {\frac{{4{x^4}}}{4}} \right)_3^3 \cr & = \left( {{x^4}} \right)_0^3 \cr & = {\left( 3 \right)^4} - {\left( 0 \right)^4} \cr & {\text{simplifying}} \cr & = 81 \cr & \int_a^b {f\left( x \right)} dx \ne 1 \cr & \cr & {\text{Condition 2 is not satisfied}}{\text{, so}} \cr & f\left( x \right){\text{ }}is{\text{ }}not{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function \cr}