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.
-$$f ( x ) = \frac { 1 } { 7 } x \text { over } [ 4 , k ]$$
A) 30
B) $3 \sqrt { 2 }$
C) $\sqrt { 30 }$
D) $\sqrt { 23 }$

Question

Quizplus · Accepted Answer

For a function to be a probability density function (pdf), the total area under the curve over the specified interval must equal 1. The function given is $f(x) = \frac{1}{7}x$, and it is defined over the interval $[4, k]$. To find the value of $k$, we integrate $f(x)$ from 4 to $k$ and set the integral equal to 1.$$\int_{4}^{k} \frac{1}{7}x \, dx = 1$$Solving the integral:$$\frac{1}{7} \cdot \frac{x^2}{2} \Big|_{4}^{k} = 1$$$$\frac{1}{14} \cdot (k^2 - 16) = 1$$$$k^2 - 16 = 14$$$$k^2 = 30$$$$k = \sqrt{30}$$Therefore, the correct answer is $k = \sqrt{30}$, which is option C.

Find the Value of the Constant K So That the Given