Let $X$ be a continuous random variable taking values in $[0, \sqrt{3}]$. Let the probability density function of $X$ be

$$f_{X}(x)=\frac{c}{1+x^{2}}, \quad \text { for } x \in[0, \sqrt{3}]$$

where $c$ is a constant.

Find the value of $c$ and calculate the mean, variance and median of $X$.

[Recall that the median of $X$ is the number $m$ such that $\left.\mathbb{P}(X \leqslant m)=\frac{1}{2} .\right]$