How to calculate the median of a continuous random variable

Question

$X$ is a continuous random variable with probability density function $f(x)= \dfrac{2x}{15}$ where $1≤x≤4$.

What is the median of X?

Answer 1

Hint: To find the median, you want to find $c$ such that $\mathbb{P}(1 \le X \le c) = \mathbb{P}(c \le X \le 4)$. That's just the definition of the median: it's the number $c$ for which the probabilities on both of its sides are the same.

The integral you get here shouldn't be hard to carry out.

Answer 2

Hint: You have to find $u$ such that $P(-\infty < x < u) = 0.5$.