Let X denote the number of hours a person watches television during a randomly selected day. The probability that X can take the values x; has the following form, where k is some unknown constant.

$P(X = x;) =\left\{\begin{matrix}0.2,&if\,x_i=0\\Kx_i,& if\, x_i = 1\, or\, 2\\k (5-x_i),&if\, x_i = 3\\0,&otherwise\end{matrix}\right.$

Find the value of k.

$\frac{4}{25}$

The correct answer is Option (1) → $\frac{4}{25}$

From the given information, we find that the probability distribution of X is

We know that $Σp_i = 1$

$⇒ 0.2+k+2k + 2k = 1$

$⇒ 5k=0.8⇒k=\frac{4}{25}$