Find the constant k such that the function $f ( x ) = k x$ is a probability density function over the interval $[ 3,8 ]$ .
A)5
B) $\frac { 2 } { 55 }$
C) $\frac { 2 } { 5 }$
D) $\frac { 55 } { 2 }$
E) $\frac { 5 } { 2 }$

Question

Quizplus · Accepted Answer

For a function $f(x) = kx$ to be a probability density function over the interval $[3,8]$, it must satisfy the condition $\int_{3}^{8} kx \, dx = 1$. Solving the integral gives $\frac{1}{2}kx^2$ evaluated from 3 to 8, which simplifies to $\frac{1}{2}k(8^2 - 3^2) = \frac{1}{2}k(64 - 9) = \frac{1}{2}k(55)$. Setting this equal to 1 and solving for $k$ gives $k = \frac{2}{55}$.

Find the Constant K Such That the Function f(x)=kxf ( x ) = k xf(x)=kx

Find the Constant K Such That the Function $f ( x ) = k x$