Finding the Mode of a Continuous Random Variable

The mode of a continuous random variable $X$ with probability density function $f$ is the value $x^*$ in the support of $X$ at which $f$ attains its maximum:

$$x^* = \arg\max_{x} f(x).$$

For a smooth PDF on an interval, we find the mode by setting $f'(x)=0$ and identifying which critical point yields the largest value of $f$.

Illustration. Let $X$ have PDF $f(x) = 6x(1-x)$ on $[0,1]$. Then

$$f'(x) = 6 - 12x.$$

Setting $f'(x) = 0$ gives $x = \dfrac{1}{2}$. Since $f''(x) = -12 < 0$, this critical point is a maximum. The mode of $X$ is $x^* = \dfrac{1}{2}$.