## Calculus with Applications (10th Edition)

$$f\left( x \right){\text{ }}is{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function$$
\eqalign{ & f\left( x \right) = 0.4;\,\,\,\,\,\,\,\left[ {4,6.5} \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 & f\left( x \right) = 0.4{\text{ is positive for all real number }}x \cr & \cr & 2{\text{ condition}}:\int_a^b {f\left( x \right)} dx = 1.{\text{ then}} \cr & \int_4^{6.5} {0.4} dx = 1 \cr & {\text{integrating}} \cr & = \left( {0.4x} \right)_4^{6.5} \cr & = 0.4\left( {6.5} \right) - 0.4\left( 4 \right) \cr & {\text{simplifying}} \cr & = 2.6 - 1.6 \cr & = 1 \cr & \cr & {\text{The condition 1 and 2 are verified}}{\text{, then }} \cr & f\left( x \right){\text{ }}is{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function \cr}