For the given probability density function, over the stated interval, find the requested value. $f(x)=\frac{1}{4} x$, over the interval $[1,4]$. Find $E(x)$
A. $\frac{3}{2}$
B. $\frac{16}{3}$
C. $\frac{21}{4}$
D. $\frac{31}{6}$

Step 1 ：The expected value $E(x)$ of a continuous random variable is given by the integral of $x$ times the probability density function $f(x)$ over the entire range of $x$. In this case, the range of $x$ is from 1 to 4, and $f(x) = \frac{1}{4}x$. So, we have $E(x) = \int_{1}^{4} x \cdot f(x) dx$.

Step 2 ：Substitute $f(x)$ into the integral, we get $E(x) = \int_{1}^{4} x \cdot \frac{1}{4}x dx = \frac{1}{4} \int_{1}^{4} x^2 dx$.

Step 3 ：The integral of $x^2$ from 1 to 4 is $\frac{1}{3}x^3$ evaluated from 1 to 4, which is $\frac{1}{3}(4^3 - 1^3)$.

Step 4 ：Calculate the value, we get $E(x) = \frac{1}{4} \cdot \frac{1}{3}(4^3 - 1^3) = \frac{1}{4} \cdot \frac{1}{3}(64 - 1) = \frac{1}{4} \cdot \frac{1}{3} \cdot 63 = \frac{21}{4}$.

Step 5 ：So, the expected value $E(x)$ is $\frac{21}{4}$.