Find the value of the constant k so that the given function in a probability density function for a random variable over the
specified interval.
-$f ( x ) = k ( 19 - x )$ over $[ 0,19 ]$
A) $\frac { 1 } { 361 }$
B) 361
C) $\frac { 2 } { 361 }$
D) 19

Question

Quizplus · Accepted Answer

For a function to be a probability density function (pdf) over a certain interval, the integral of the function over that interval must equal 1. Given $f(x) = k(19 - x)$ over the interval $[0, 19]$, we find $k$ by solving the integral:$$\int_{0}^{19} k(19 - x) dx = 1$$Solving this integral:$$k \left[ 19x - \frac{x^2}{2} \right]_{0}^{19} = 1$$$$k \left[ 19(19) - \frac{19^2}{2} \right] = 1$$$$k \left[ 361 - \frac{361}{2} \right] = 1$$$$k \left[ \frac{361}{2} \right] = 1$$$$k = \frac{2}{361}$$Therefore, the correct value of $k$ that makes $f(x)$ a probability density function over the interval $[0, 19]$ is $\frac{2}{361}$.

Find the Value of the Constant K So That the Given