Question and Answers Forum

All Questions      Topic List

Probability and Statistics Questions

Previous in All Question      Next in All Question      

Previous in Probability and Statistics      Next in Probability and Statistics      

Question Number 50855 by peter frank last updated on 21/Dec/18

$$F\mathrm{or}therandomvariable$$$$xshowthat$$$$a)Var\left(x\right)=E^2\left(x\right)-{\left[E\left(x\right)\right]}^2$$$$b)Var(ax+b)=a^{2}Var\left(x\right)$$$$$$

Answered by peter frank last updated on 21/Dec/18

$$var\left(x\right)=\frac{\mathrm{\Sigma }f{(x-\stackrel{-}{x})}^2}{N}$$$$var\left(x\right)=\frac{\mathrm{\Sigma }f[x^2-2x\stackrel{-}{x}+{\left(\stackrel{-}{x}\right)}^2]}{N}$$$$var\left(x\right)=\frac{\mathrm{\Sigma }{fx}^2}{N}-2\stackrel{-}{x}\frac{\mathrm{\Sigma }fx}{N}+{\left(\stackrel{-}{x}\right)}^2$$$$var\left(x\right)=\frac{\mathrm{\Sigma }{fx}^2}{N}-2\stackrel{-}{x}\stackrel{-}{x}+{\left(\stackrel{-}{x}\right)}^2$$$$var\left(x\right)=\frac{\mathrm{\Sigma }{fx}^2}{N}-2{\left(\stackrel{-}{x}\right)}^2+{\left(\stackrel{-}{x}\right)}^2$$$$var\left(x\right)=\frac{\mathrm{\Sigma }{fx}^2}{N}-{\left(\stackrel{-}{x}\right)}^2$$$$\stackrel{-}{x}=E\left(x\right)$$$$var\left(x\right)=E\left(x^2\right)-{\left[E\left(x\right)\right]}^2$$$$shown$$$$$$$$fromvar\left(x\right)=E\left(x^2\right)-{\left[E\left(x\right)\right]}^2$$$$var(ax+b)=E{(ax+b)}^2-{\left[E(ax+b)\right]}^2$$$$\frac{\mathrm{\Sigma }f(a^2{x}^2+2abx+b^2)}{N}-[a^2{\left[E\left(x\right)\right]}^2-2abE\left(x\right)+b^2]$$$$simplify$$$$a^{2}E\left(x^2\right)-a^2{\left[E\left(x\right)\right]}^2$$$$a^{2}var\left(x\right)$$$$shown$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com