x	$$ - $$2	$$ - $$1	3	4	6
P(X = x)	$${1 \over 5}$$	a	$${1 \over 3}$$	$${1 \over 5}$$	b

$$\overline X $$ = 2.3

$$-$$a + 6b = $${9 \over {10}}$$ ..... (1)

$$\sum {{P_i} = {1 \over 5} + a + {1 \over 3} + {1 \over 5} + b = 1} $$

$$a + b = {4 \over {15}}$$ .... (2)

From equation (1) and (2)

$$a = {1 \over {10}},b = {1 \over 6}$$

$${\sigma ^2} = \sum {{p_i}x_i^2 - {{(\overline X )}^2}} $$

$${1 \over 5}(4) + a(1) + {1 \over 3}(9) + {1 \over 5}(16) + b(36) - {(2.3)^2}$$

$$ = {4 \over 5} + a + 3 + {{16} \over 5} + 36b - {(2.3)^2}$$

$$ = 4 + a + 3 + 36b - {(2.3)^2}$$

$$ = 7 + a + 36b - {(2.3)^2}$$

$$ = 7 + {1 \over {10}} + 6 - {(2.3)^2}$$

$$ = 13 + {1 \over {10}} - {\left( {{{23} \over {10}}} \right)^2}$$

$$ = {{131} \over {10}} - {\left( {{{23} \over {10}}} \right)^2}$$

$$ = {{1310 - {{(23)}^2}} \over {100}}$$

$$ = {{1310 - 529} \over {100}}$$

$${\sigma ^2} = {{781} \over {100}}$$

$$100{\sigma ^2} = 781$$