Let $X$ be a random variable that represents the length of the diameter of a circle, we know that $X:\mathcal{U}(5,7)$. Find expected value of circle area.
As far as i know, first, we need density function for this random variable. We know that this variable is uniformly distributed so $f(x)=\frac{1}{2}$. Now, we know that area of a circle is $A=\pi r^2$ where $r$ is radius. As far as i know, formula for expected value is: $$E(X)=\int_{-\infty}^{\infty}xf(x)dx$$ I have that $f(x)=\frac{1}{2}$. How am i supposed to include formula for area of circle into all of this? Any help appreciated!
Two observations:
$\qquad(1)$: $f(x)$ is not $\frac12$, but rather $\frac12 \chi_{[5,7]}$. In other words, $f$ is $0$ outside $[5,7]$.
$\qquad(2)$: You wish to calculate the expectation of the area, which is $\pi\, (X/2)^2$. Hence, you wish to calculate
$$\mathbb E\left(\pi {(X/2)}^2\right) = \frac\pi4\,\mathbb E\left( X^2\right).$$
Can you conclude?