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Homemedian of a distribution
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median of a distribution
Given a probability distribution (density) function $f_{X}(x)$ on $\Omega$ over a random variable $X$, with the associated probability measure $P$, a median $m$ of $f_{X}$ is a real number such that
1. $P(X\leq m)\geq\frac{1}{2},$
2. $P(X\geq m)\geq\frac{1}{2}.$
The median is also known as the $50^{{\text{th}}}$percentile or the second quartile.
Examples:

An example from a discrete distribution. Let $\Omega=\mathbb{R}$. Suppose the random variable $X$ has the following distribution: $P(X=0)=0.99$ and $P(X=1000)=0.01$. Then we can easily see the median is 0.

The median of a normal distribution (with mean $\mu$ and variance $\sigma^{2}$) is $\mu$. In fact, for a normal distribution, mean = median = mode.

The median of a uniform distribution in the interval $[a,b]$ is $(a+b)/2$.

The median of a Cauchy distribution with location parameter t and scale parameter s is the location parameter.

The median of an exponential distribution with location parameter $\mu$ and scale parameter $\beta$ is the scale parameter times the natural log of 2, $\beta\operatorname{ln}2$.

The median of a Weibull distribution with shape parameter $\gamma$, location parameter $\mu$, and scale parameter $\alpha$ is $\alpha(\operatorname{ln}2)^{{1/\gamma}}+\mu$.
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