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.
-$$f ( x ) = \frac { k } { x } \text { over } [ 1,17 ]$$
A) $\frac { 1 } { \ln 17 }$
B) $\ln 17$
C) $1 - \ln 17$
D) $\frac { 2 } { \ln 17 }$

Question

Quizplus · Accepted Answer

A probability density function must satisfy two conditions:

1. The area under the curve of the function over the given interval must be equal to 1 (since the probability of any outcome occurring is 1).

2. The function must be non-negative over the given interval.

We can first set up the integral to find the value of k:

\begin{align*}
1 &=\int_{1}^{17} \frac{k}{x} \ dx\
&=k\ln|x| \bigg|_{1}^{17}\
&=k\ln(17) - k\ln(1)\
&=k\ln(17)
\end{align*}

Since $k$ is the only unknown, we can solve for it by dividing both sides by $\ln(17)$, giving us $k = \frac{1}{\ln(17)}$. Therefore, the answer is choice A.

Find the Value of the Constant K So That the Given