If you will let me assume that the probability density function (pdf) is absolutely continuous over its support then the median is given as the integral from -inf to the median of the pdf over that support = 1/2.
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how do i find the median of a continuous probability distribution
The formula is: median of lognormal = exp(u)
Yes, mode equals median in a normal distribution.
Yes it is. The normal distribution is symmetrical around the mode. Therefore the median has to be the same :)
Mean, median, and mode are all equal in a normal distribution.