Q.6.33 - A First Course in Probability

Question

Let $X1, ... , Xn$ be independent and identically distributed random variables having distribution function F and density f. The quantity $M K \left[X\right(1) + X(n\left)\right]/2$, 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('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('

Step-by-Step Solution

Answer

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

(b) $$\begin{gathered} Probability: \\ P (Y\ge 2)=0.016 \end{gathered}$$

Step 1: Introduction

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

Step 2: Explanation

$$\begin{gathered} The expected number of typographical error is 0.2 \\ such that \\ np=0.2 \\ The number of letter is assumed to be very, very high. \\ Since the distribution of number of errors is Binomial, \\ With parameters n and p. \\ We have \\ The expected number, \\ np=0.2 \\ But \\ n is unknown. \\ Thus, \\ p cannot be determined. \\ Then Poisson approximation can be used. \\ With parameter \\ \lambda =np=0.2 \\ Let Y be the number of errors having approx. Pois (0.2) distribution. \\ P(Y=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!} \\ Thus, \\ P(Y=0)=\frac{0.2^0{e}^{-0.2}}{0!}=e^{-0.2}=0.82 \end{gathered}$$

Step3 : Given Information

$$\begin{gathered} Expected number of typographical errors is 0.2 . \\ Such that \\ n p=0.2 \\ The number of letter is assumed to be very, very high. \\ Since the distribution of number of errors is Binomial, \\ With parameters nand p. \\ We have \\ The expected number, \\ n p=0.2 \\ But \\ n is unknown. \\ Thus, p cannot be determined. \\ Then \\ Poisson approximation can be used. \\ With parameter \\ \lambda =np=0.2 \\ Let Y be the number of errors having approx. \\ Pois (0.2) distribution. \\ P(Y=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!} \\ Thus, \\ P(Y \backslash geq 2) \&=1-P(Y=0)-P(Y=1) \\ =1-e^{-\lambda }-\lambda e^{-\lambda } \\ =1-e^{-0.2}-0.2e^{-0.2} \\ =1-0.82-0.164 \\ =0.016 \end{gathered}$$