Let X denote the number of hours a student studies on a selected day. The probability distribution of X is given by (where $k$ is some unknown constant)

$P(X = x)=\left\{\begin{matrix}0.5;&if\,x_i=0\\kx_i;&if\,x_i=1\\k(4-x_i);&x_i=2\,or\,3\\0;&otherwise\end{matrix}\right.$ Then the value of $k$ is

$k=\frac{1}{8}$

The correct answer is Option (3) → $k=\frac{1}{8}$ **

Given probability distribution:

$P(X=0)=0.5$
$P(X=1)=k\cdot 1 = k$
$P(X=2)=k(4-2)=2k$
$P(X=3)=k(4-3)=k$

Total probability = 1:

$0.5 + k + 2k + k = 1$

$0.5 + 4k = 1$

$4k = 0.5$

$k = \frac{0.5}{4} = \frac{1}{8}$

$k = \frac{1}{8}$