The probability distribution of X is : x 0 1 2 3 4 P(X = x) 0.1 2k k…

x	0	1	2	3	4
P(X = x)	0.1	2k	k	k	2k

∑P(X_i) = 1

⇒ 0.1 + 2k + k + k + 2k = 1

⇒ 6k = 0.7

so $k = \frac{0.9}{6} = \frac{9}{60} = \frac{3}{20}$

so var(X) = E(X²) - E²(X)

= 0² × 0.1 + 1² × 2k + 2² × k + 3² × k + 4² × 2k - (0 × 0.1 + 1 × 2k + 2 × k + 3 × k + 4 × 2k)²

= 2k + 4k + 9k + 32k - (2k + 2k + 3k + 8k)²

= 47 - (15k)²

= 47k - 225k²

= $47 × \frac{3}{20} - 225 × \frac{9}{400}$

= $\frac{141}{20} - \frac{81}{16}$

= $\frac{564 - 405}{80}$

= $\frac{159}{80}$