Question

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Answer 1

$\displaystyle \\\\F(x)= \begin{cases} 0 ~~~~~~~~~~~~~~\text{ (x}\leq 0), \\ \frac{1}{4}x^{2} +\frac{3}{4}x ~~~~ (0<\text{x} \leq 1), \\ 1\quad~~~~~~~~~~~~(\text{x}>1). \end{cases}$

$\displaystyle f(x)=\begin{cases} \frac{x}{2}+\frac{3}{4}, ~~~~(x\in[0;1]) \\ 0,~~~~~~~~~~(x\notin[0;1]) \end{cases}$

$\displaystyle M(x) = \int\limits^1_0 {x*f(x)} \, dx =\int\limits^1_0 { \frac{x^{2}}{2}+\frac{3x}{4} } \, dx =\frac{x^{3} }{6} +\frac{3x^{2} }{8} \quad\bigg|^1_0=\frac{1}{6} +\frac{3}{8} =\frac{13}{24}\approx0.542$

$\displaystyle D(x) = \int\limits^1_0 {x^{2} *f(x)} \, dx =\int\limits^1_0 {\frac{x^{3} }{2} } \, +\frac{3x^{2} }{4} dx =\frac{x^{4} }{8} +\frac{x^{3} }{4} \quad~~\bigg|^1_0=\frac{1}{8} +\frac{1}{4} =\frac{3}{8}=0.375$

$\displaystyle \sigma(x)=\sqrt{D(x)}=\sqrt{\frac{3}{8} } \approx0.612$