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Question Number 201972 by Lekhraj last updated on 17/Dec/23

Answered by mr W last updated on 18/Dec/23

$${\mathbf{\Phi }}_{⩽\mathit{x}}=\frac{1}{2}[1+\mathit{e}\mathit{r}\mathit{f}\left(\frac{\mathit{x}-\mathit{\mu }}{\sqrt{2}\mathit{\sigma }}\right)]$$$$\mathit{w}\mathit{i}\mathit{t}\mathit{h}\mathit{e}\mathit{r}\mathit{f}\left(\mathit{x}\right)=\frac{2}{\sqrt{\mathit{\pi }}}{\int }_0^{\mathit{x}}{\mathit{e}}^{-{\mathit{t}}^2}\mathit{d}\mathit{t}$$$$$$$$\left(i\right)\left(a\right)$$$$\frac{1}{2}[1+erf\left(\frac{45-60}{10\sqrt{2}}\right)]$$$$=\frac{1}{2}(1-0.886)\approx 6\mathrm{\%}$$$$\left(i\right)\left(b\right)$$$$\frac{1}{2}[erf\left(\frac{75-60}{10\sqrt{2}}\right)-erf\left(\frac{50-60}{10\sqrt{2}}\right)]$$$$=\frac{1}{2}(0.866-(-0.683))\approx 77\mathrm{\%}$$$$\left(ii\right)$$$$\frac{1}{2}[1+erf\left(\frac{x-60}{10\sqrt{2}}\right)]=0.9$$$$erf\left(\frac{x-60}{10\sqrt{2}}\right)=0.8$$$$\frac{x-60}{10\sqrt{2}}={erf}^{-1}(0.8)=0.9062$$$$\Rightarrow x=60+0.9062\times 10\sqrt{2}\approx 73\left(\mathrm{\%}\right)$$

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