Probability

A random variable $X$ has a probability density function $\;$ $f \left(x\right) = \begin{cases} k, & 0 < x < 2 \pi \\ & \\ 0, & \text{ elsewhere} \end{cases}$
Find:

1. $k$
2. $P \left(0 < x < \dfrac{\pi}{2}\right)$
3. $P \left(\dfrac{\pi}{2} < x < \dfrac{3 \pi}{2}\right)$

1. $\because \;$ $f \left(x\right)$ is a probability density function, $\int \limits_{-\infty}^{+ \infty} f \left(x\right) dx = 1$
   
   $\therefore \;$ $\int \limits_{0}^{2 \pi} f \left(x\right) dx = 1$
   
   i.e. $\;$ $\int \limits_{0}^{2 \pi} k \; dx = 1$
   
   i.e. $\;$ $k \times \left[x\right]_{0}^{2 \pi} = 1$
   
   $\implies$ $k = \dfrac{1}{2 \pi}$
   
   
2. $P \left(0 < x < \dfrac{\pi}{2}\right) = \int \limits_{0}^{\pi / 2} f \left(x\right) dx$
   
   Now, $f \left(x\right) = \begin{cases} \dfrac{1}{2 \pi}, & 0 < x < 2 \pi \\ & \\ 0, & \text{ elsewhere} \end{cases}$
   
   $\begin{aligned} \therefore \; P \left(0 < x < \dfrac{\pi}{2}\right) & = \dfrac{1}{2 \pi} \int \limits_{0}^{\pi / 2} dx \\\\ & = \dfrac{1}{2 \pi} \times \left[x\right]_{0}^{\pi / 2} \\\\ & = \dfrac{1}{2 \pi} \times \dfrac{\pi}{2} = \dfrac{1}{4} \end{aligned}$
   
   
3. $P \left(\dfrac{\pi}{2} < x < \dfrac{3 \pi}{2}\right) = \int \limits_{\pi / 2}^{3 \pi / 2} f \left(x\right) dx$
   
   $\begin{aligned} \therefore \; P \left(\dfrac{\pi}{2} < x < \dfrac{3 \pi}{2}\right) & = \dfrac{1}{2 \pi} \int \limits_{\pi / 2}^{3 \pi / 2} dx \\\\ & = \dfrac{1}{2 \pi} \times \left[x\right]_{\pi / 2}^{3 \pi / 2} \\\\ & = \dfrac{1}{2 \pi} \times \left(\dfrac{3 \pi}{2} - \dfrac{\pi}{2}\right) = \dfrac{1}{2} \end{aligned}$