# Definition of a median of a random variable

A median of a random variable $X$ is defined as a number $m \in \mathbb{R}$ such that $P(X \leq m) \geq \frac{1}{2}$ and $P(X \geq m) \geq \frac{1}{2}$.

Is it equivalent to say: $m$ is a median of $X$ iff $P(X < m) = P(X > m)$?

Is it true that $m$ is a median of $X$ iff $P(X<m)\leq \frac{1}{2}$ and $P(X>m)\leq \frac{1}{2}$? (This is easy. It is true.)

Thanks and regards!

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Suppose $X$ takes on values $0$, $1$, and $2$ with probabilities $0.2, 0.5, 0.3$ respectively. What is the median $m$? What are $P\{X < m\}$ and $P\{X > m\}$? – Dilip Sarwate Nov 26 '11 at 1:55

No. Suppose that $X$ takes on values $0$, $1$, and $2$ with probabilities $\dfrac{1}{3}$, $\dfrac{1}{4}$, and $\dfrac{5}{12}$ respectively.
Then $1$ is a median of $X$ according to the first definition, but not according to the second.
Thanks! Is it true that $m$ is a median of $X$ iff $P(X<m)\leq 0.5$ and $P(X>m)\leq 0.5$? (Edit: This is easy. It is true.) – Tim Nov 26 '11 at 2:04