I am not sure how to solve the following problem:
The probability density function of the Rayleigh distribution is,
$\ f(x;α) = \frac{x}{α^2} e^\frac{-x^2}{2α^2}, x ≥ 0, $
where α is the scale parameter of the distribution. Find the median of the Rayleigh distribution.
I need to derive the median of the distribution, but do not know how to do so. Thoughts?
Best Answer
The Rayleigh distribution has cumulative distribution function (CDF) $F_X(x) = 1-\text{e}^{\frac{-x^2}{2\alpha^2}}$.
Denote the median $q_{50}$.
Starting with the CDF...
$$\begin{align} 1-\text{e}^{\frac{-q_{50}^2}{2\alpha^2}} &= 0.5 \\ \text{e}^{\frac{-q_{50}^2}{2\alpha^2}} &= 0.5 \\ \frac{-q_{50}^2}{2\alpha^2} &= \text{ln}(0.5) \\ -q_{50}^2 &= 2\alpha^2 \text{ln}(0.5) \\ \\ q_{50} &=\alpha \sqrt{-2 \text{ln}(0.5)} \\ &= \alpha \sqrt{2\text{ln}(2)} \quad \quad \square \end{align}$$
See here or here for general quantiles.
Update: Based on comments amounting to "can I do this from the PDF," yes, it is possible but requires a little more effort (integration).
You can solve $\int_0^{q_{50}} f_X(x)dx = 0.5$ for $q_{50}$.
$$\begin{align}\int_0^{q_{50}} f_X(x)dx &= 0.5 \\ \int_0^{q_{50}} \frac{x}{\alpha^2}\text{e}^{-x^2/(2\alpha^2)} dx&= 0.5 \\ 1-\text{e}^{\frac{-q_{50}^2}{2\alpha^2}} &= 0.5 \\ &\text{Continue using CDF approach above} \end{align}$$