Let X denote the number of hours a person watches television during a randomly selected day. The probability that X can take the values $x_i$ 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 variance and standard deviation of random variable X.

Variance = 1.22, Standard Deviation = 1.1

The correct answer is Option (4) → Variance = 1.22, Standard Deviation = 1.1

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

We know that $Σp_i = 1$

$⇒ 0.2+k+2k + 2k = 1$

$⇒ 5k=0.8⇒k=\frac{4}{25}$

We construct the following table:

$E(X) = Σp_ix_i=\frac{44}{25}= 1.76$

Variance $σ^2 = Σp_i{x_i}^2 - (Σp_ix_i)^2$

$=\frac{44}{25}-(\frac{44}{25})^2=\frac{108}{25}-\frac{1936}{625}=\frac{764}{625}= 1.22$

and standard deviation $σ =\sqrt{Variance}=\sqrt{1.22} = 1.1$