C..2.0.4 4

(1)

$$1=\int_{-\infty}^\infty f(x) \Dx=\int_0^1 ax(1-x)\Dx =a\int_0^1x(1-x)\Dx=a\left(\frac{1}{2}-\frac{1}{3}\right)=\frac{a}{6}.$$

ゆえに $a=6$.
(2)

   平均$$=\int_{-\infty}^\infty xf(x) \Dx =\int_0^1x\cdot 6x(1-x)\Dx =6\int_0^1(x^2-x^3)\Dx =6\cdot\left(\frac{1}{3}-\frac{1}{4}\right) =\frac{1}{2},$$

$$\int_{-\infty}^\infty x^2f(x) \Dx=\int_0^1x^2\cdot 6x(1-x)\Dx = 6\int_0^1(x^3-x^4)\Dx=6\cdot\left(\frac{1}{4}-\frac{1}{5}\right) =\frac{3}{10}$$

であるから

   分散$$=\frac{3}{10}-\left(\frac{1}{2}\right)^2=\frac{1}{20}.$$