exercise:C16e9ef582: Difference between revisions
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Suppose <math>F(x)</math> is a continuous cumulative probability distribution function with <math>\lim_{x\rightarrow 1}F(x)=1</math> and <math>F(x)>0</math> for all <math>x</math>. For which of the following <math>g(x)</math> is <math>F(g(x))</math> also a cumulative probability distribution function? | Suppose <math>F(x)</math> is a continuous cumulative probability distribution function with <math>\lim_{x\rightarrow 1}F(x)=1</math> and <math>F(x)>0</math> for all <math>x</math>. For which of the following <math>g(x)</math> is <math>F(g(x))</math> also a cumulative probability distribution function? | ||
< | <ul class="mw-excansopts"> | ||
<li><math>x^2</math></li> | <li><math>x^2</math></li> | ||
<li><math>\sqrt{|x| + 1} </math></li> | <li><math>\sqrt{|x| + 1} </math></li> | ||
| Line 7: | Line 7: | ||
<li><math>(1 + e^{-x})^{-1}</math></li> | <li><math>(1 + e^{-x})^{-1}</math></li> | ||
<li><math>1-\ln(1 + e^{-x})</math></li> | <li><math>1-\ln(1 + e^{-x})</math></li> | ||
</ | </ul> | ||
Latest revision as of 12:43, 18 March 2024
Suppose [math]F(x)[/math] is a continuous cumulative probability distribution function with [math]\lim_{x\rightarrow 1}F(x)=1[/math] and [math]F(x)\gt0[/math] for all [math]x[/math]. For which of the following [math]g(x)[/math] is [math]F(g(x))[/math] also a cumulative probability distribution function?
- [math]x^2[/math]
- [math]\sqrt{|x| + 1} [/math]
- [math]e^{-x}[/math]
- [math](1 + e^{-x})^{-1}[/math]
- [math]1-\ln(1 + e^{-x})[/math]