The probability distribution of the random variable X is given by

X	0	1	2	3
P(X)	0.2	k	2k	2k

The variance of the random variable X is

The sum of all probabilities in a probability distribution must equal 1.

$P(X=0) + P(X=1) + P(X=2) + P(X=3) = 1$

$0.2 + k + 2k + 2k = 1$

$0.2 + 5k = 1$

$5k = 0.8$

$k = 0.16 = \frac{4}{25}$

---

The complete probability distribution is:

X	0	1	2	3
P(X)	$\frac{1}{5}$	$\frac{4}{25}$	$\frac{8}{25}$	$\frac{8}{25}$

---

The mean is given by $E(X) = \sum x \cdot P(x)$

$E(X) = 0 \times \frac{5}{25} + 1 \times \frac{4}{25} + 2 \times \frac{8}{25} + 3 \times \frac{8}{25}$

$= 0 + \frac{4}{25} + \frac{16}{25} + \frac{24}{25}$

$= \frac{44}{25}$

---

$E(X^2) = \sum x^2 \cdot P(x)$

$E(X^2) = 0^2 \times \frac{5}{25} + 1^2 \times \frac{4}{25} + 2^2 \times \frac{8}{25} + 3^2 \times \frac{8}{25}$

$= 0 + \frac{4}{25} + 4 \times \frac{8}{25} + 9 \times \frac{8}{25}$

$= \frac{4}{25} + \frac{32}{25} + \frac{72}{25}$

$= \frac{108}{25}$

---

The variance is given by $\text{Var}(X) = E(X^2) - [E(X)]^2$

$\text{Var}(X) = \frac{108}{25} - \left(\frac{44}{25}\right)^2$

$= \frac{108}{25} - \frac{1936}{625}$

$= \frac{2700}{625} - \frac{1936}{625}$

$= \frac{764}{625}$

Therefore, the variance of the random variable X is $\frac{764}{625}$.