Question-19

median
cdf
continuous random variable
DA-2025

Let \(X\) be a continuous random variable whose cumulative distribution function (CDF) \(F_{X}( x)\), for some \(t\), is given as follows:

\[ F_{X}( x) =\begin{cases} 0, & x\leqslant t\\ \cfrac{x-t}{4-t} , & t\leqslant x\leqslant 4\\ 1, & x\geqslant 4 \end{cases} \]

If the median of \(X\) is \(3\), then what is the value of \(t\)?

The median is that value of \(x\) for which \(F_{X}( x) =0.5\). The median here is \(3\):

\[ \begin{array}{ c r l } & \cfrac{3-t}{4-t} & =\cfrac{1}{2}\\ \Longrightarrow & 6-2t & =4-t\\ \Longrightarrow & t & =2 \end{array} \]