286. The random variable $X$ in the interval $(2,4)$ is given by the probability density function $f(x)=-(3 / 4) x^{2}+(9 / 2) x-6$; outside this interval, $f(x)=0$. Find the mode, the expected value, and the median of the variable $X$.
Solution. Represent the density function as $f(x)=$ $=-(3 / 4)(x-3)^{2}+3 / 4$. From this, it is clear that the density function reaches its maximum at $x=3$; therefore, $M_{0}(X)=3$. (Of course, the maximum could have been found using methods of differential calculus.)

The distribution curve is symmetric with respect to the line $x=3$, hence $M(X)=3$ and $M_{e}(X)=3$.