Determine random variable such as Med[X]=a and E[X]=b.

Question

Let $a<b$ (a,b real numbers). Determine a random variable $X$ such as $Med[X]=a$ and $E[X]=b$.

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Hmmm how does one find $E[X]$ and $Med[X]$ if we don't have the probability distribution? The only way if X=const but then the median and E-value would be equal.

Answer 1

Let $Y$ be a random variable with median $0$ and mean $c=b-a$. Then setting $X=Y+a$ will give us the desired result.

To construct a $Y$ that works, let $Y$ have density function $\frac{1}{2}$ in the interval $[-1,0)$, and $\frac{1}{2p}$ in the interval $[0,p]$. Then the median of $Y$ is $0$. The mean of $Y$ can be calculated by integrating, or more simply by observing that it is equal to $\frac{1}{2}\left(-\frac{1}{2}+\frac{p}{2}\right)$.

Set this equal to $c$. We get $p=4c+1$.

Answer 2

Let $X$ be $a$, 51% of the time, and some other number $c$, 49% of the time. The median will be $a$; and if you vary $c$ you can make the expected value be whatever you want, namely $b$.