Find the constant $k$ such that the function $f ( x ) = k e ^ { - \frac { 9 } { 7 } x }$ is a probability density function over the interval $[ 0 , \infty ]$ .
A)1
B) $\frac { 7 } { 9 }$
C) $- \frac { 7 } { 9 }$
D) $- \frac { 9 } { 7 }$
E) $\frac { 9 } { 7 }$

Question

Quizplus · Accepted Answer

The integral of a probability density function over its support must be equal to 1. So,$$\int_0^\infty k e^{-\frac{9}{7}x} \ dx = 1$$$$\Rightarrow k\left[-\frac{7}{9}e^{-\frac{9}{7}x}\right]_0^\infty = 1$$$$\Rightarrow k\left(0-\left(-\frac{7}{9}\right)\right) = 1$$$$\Rightarrow k = \frac{9}{7}$$

Find the Constant kkk Such That the Function f(x)=ke−97xf ( x ) = k e ^ { - \frac { 9 } { 7 } x }f(x)=ke−79​x

Find the Constant $k$ Such That the Function $f ( x ) = k e ^ { - \frac { 9 } { 7 } x }$