## Calculus with Applications (10th Edition)

$$f\left( x \right){\text{ }}is{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function$$
\eqalign{ & f\left( x \right) = \frac{{{x^2}}}{{21}};\,\,\,\,\,\,\,\left[ {1,4} \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 & \frac{{{x^2}}}{{21}}{\text{ is positive for all real number }}x,{\text{ then }}f\left( x \right) \geqslant 0{\text{ for the given interval}} \cr & \cr & 2{\text{ condition}}:\int_a^b {f\left( x \right)} dx = 1.{\text{ then}} \cr & \int_1^4 {\left( {\frac{{{x^2}}}{{21}}} \right)} dx \cr & {\text{integrating}} \cr & = \left( {\frac{{{x^3}}}{{3\left( {21} \right)}}} \right)_1^4 \cr & = \frac{1}{{63}}\left( {{{\left( 4 \right)}^3} - {{\left( 1 \right)}^3}} \right) \cr & {\text{simplifying}} \cr & = \frac{1}{{63}}\left( {64 - 1} \right) \cr & = 1 \cr & \cr & {\text{The conditions are verified}}{\text{, then }} \cr & f\left( x \right){\text{ }}is{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function \cr}