Continuous random median

Remember that the median of any distribution is the value in the middle: 50% of the data is below it and 50% of the data is above it.

When talking about probability and distributions, the median is where the probability is 0.5.

Finding a formula

Knowing our formula for finding the probability of a continuous random variable, P(a \leq X \leq b) = \int_a^b f(x)\,dx, as well as that the median is where the probability is 0.5, we can find that:

\int_0^mf(x)\,\,dx=0.5

We can use definite integration and find the value of m - that’s the median.

Question	Answer
What is the median of any distribution?	The value in the middle: 50% of the data is below it and 50% of the data is above it.
Where is the median located in terms of probability for distributions?	The median is where the probability is 0.5.
What formula for a continuous random variable gives P(a \leq X \leq b)?	P(a \leq X \leq b) = \int_a^b f(x)\,dx
What is the defining equation for the median m of a continuous distribution?	\int_0^m f(x)\,\,dx = 0.5
How do you find the median m for a continuous random variable?	Use definite integration on \int_0^m f(x)\,dx = 0.5 and solve for m.