Question

Let $X1,...,Xn$ be independent and identically distributed random variables having distribution function F and density f. The quantity $MK\left[X\right(1)+X(n\left)\right]/2$, defined to be the average of the smallest and largest values in $X1,...,Xn$, is called the midrange of the sequence. Show that its distribution function is uncaught exception: Http Error #500

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #500') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #500') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #500') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #500') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #500') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('587f0c781406aea...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('uncaught exception: Http Error #500

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #500') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #500') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #500') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #500') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #500') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('587f0c781406aea...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

uncaught exception: Http Error #500

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #500') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #500') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #500') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #500') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #500') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('587f0c781406aea...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('.

Short answer

(a)$P(Y=0)=0.82$

(b)role="math" localid="1647327027064" $$\begin{gathered} Probability: \\ P(Y\ge 2)=0.016 \end{gathered}$$

Step-by-step solution

Introduction

Let $X1,...,Xn$be independent and identically distributed random variables having distribution function$F$and density $f$.

Explanation

$$\begin{gathered} Theexpectednumberoftypographicalerroris0.2 \\ suchthat \\ np=0.2 \\ Thenumberofletterisassumedtobevery,veryhigh. \\ SincethedistributionofnumberoferrorsisBinomial, \\ Withparametersnandp. \\ Wehave \\ Theexpectednumber, \\ np=0.2 \\ But \\ nisunknown. \\ Thus, \\ pcannotbedetermined. \\ ThenPoissonapproximationcanbeused. \\ Withparameter \\ \lambda =np=0.2 \\ LetYbethenumberoferrorshavingapprox.Pois(0.2)distribution. \\ P(Y=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!} \\ Thus, \\ P(Y=0)=\frac{0.2^0{e}^{-0.2}}{0!}=e^{-0.2}=0.82 \end{gathered}$$

Given Information

$$\begin{gathered} Expectednumberoftypographicalerrorsis0.2. \\ Suchthat \\ np=0.2 \\ Thenumberofletterisassumedtobevery,veryhigh. \\ SincethedistributionofnumberoferrorsisBinomial, \\ Withparametersnandp. \\ Wehave \\ Theexpectednumber, \\ np=0.2 \\ But \\ nisunknown. \\ Thus,pcannotbedetermined. \\ Then \\ Poissonapproximationcanbeused. \\ Withparameter \\ \lambda =np=0.2 \\ LetYbethenumberoferrorshavingapprox. \\ Pois(0.2)distribution. \\ P(Y=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!} \\ Thus, \\ P(Y\backslash geq2)\&=1-P(Y=0)-P(Y=1) \\ =1-e^{-\lambda }-\lambda e^{-\lambda } \\ =1-e^{-0.2}-0.2e^{-0.2} \\ =1-0.82-0.164 \\ =0.016 \end{gathered}$$