How to Find the Median of a Probability Distribution Function?

Question

I am interested in learning about how to find out the Median (https://en.wikipedia.org/wiki/Median) for some "generic" Probability Distribution Function (https://en.wikipedia.org/wiki/Probability_density_function).

• Suppose that for some Random Variable "X", we have have the following Probability Distribution Function : $$ P(a \leq X \leq b) = \int_{a}^{b} f(x)dx $$

• And the corresponding Cumulative Probability Distribution Function: $$ F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t)dt $$

Based on this information, I was interested in knowing if approaches like Maximum Likelihood Estimation (https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) can be used to derive a formula for the Median of this Probability Distribution Function?

As an example - suppose we consider the above "generic" Probability Distribution Function. I know that I can find out properties such as the Mean and Variance based on the Expected Values. For example:

• $$\text{mean} = \text{E}(x) = \int xf(x)dx$$
• $$ \text{variance} = \sigma^2 = \text{E}[(x - \text{E}[x])^2] = \int (x - \text{E}[x])^2f(x)dx $$

But do any such general relationships exist for the Median? How can mathematically derive such a formula?

It is pretty straightforward to calculate the Median for some data - but where exactly does the formula for the Median come from? Could this involve something like "Order Statistics" (https://en.wikipedia.org/wiki/Order_statistic)?

Thanks!

Answer 1

The median is the number $m$ s.t. \begin{align*} \int_{-\infty}^m f(x)dx = \int_m^{\infty}f(x)dx \end{align*} in other words $F(m) = \frac{1}{2}$ so you could write \begin{align*} m = F^{-1}\left(\frac{1}{2}\right) \end{align*} if you really wanted to (and if F is invertible).