Can a continuous random variable have a median?
Yes but subject to certain conditions.
The median is that value of the random variable which divides the data into two equal parts where in one half, all the values are less than the median and in the other half, all the values are greater than the median.
Translated in terms of probability, we can say this. If the range of values associated with the random variable is
#-infty#to #+infty#, and M is the median and f(x) is the pdf then we find the value of M by solving one of the two integral equations
#int_-infty^M f(x)dx#= 0.5 or
#int_M^infty f(x)dx#= 0.5
If either equation is solvable, we have the median for a continuous random variable. Otherwise, we cannot find the median.