Chapter 6: Q.6.33 (page 277)

Let $X 1, . . ., X n$ be independent and identically distributed random variables having distribution function F and density f. The quantity $M K [X (1) + X (n)] / 2$ , defined to be the average of the smallest and largest values in $X 1, . . ., X n$ , is called the midrange of the sequence. Show that its distribution function is $F M (m) = n m - q [F (2 m - x) - F (x)] n - 1 f (x) d x$ 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(' $" width="0" height="0" role="math"> F M (m) = n m - q [F (2 m - x) - F (x)] n - 1 f (x) d x$ 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(' $" width="0" height="0" role="math">$
$F M (m) = n m - q [F (2 m - x) - F (x)] n - 1 f (x) d x$ 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(' $" width="0" height="0" role="math"> F M (m) = n \int_{- \infty}^{m} [F (2 m - x) - F (x)] n - 1 f (x) d x$ .

Short Answer

Expert verified

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

(b)role="math" localid="1647327027064" $P r o b a b i l i t y : P (Y \geq 2) = 0.016$

Step by step solution

Introduction

Let $X 1, . . ., X n$ be independent and identically distributed random variables having distribution function $F$ and density $f$ .

Explanation

$T h e e x p e c t e d n u m b e r o f t y p o g r a p h i c a l e r r o r i s 0.2 s u c h t h a t n p = 0.2 T h e n u m b e r o f l e t t e r i s a s s u m e d t o b e v e r y, v e r y h i g h . S i n c e t h e d i s t r i b u t i o n o f n u m b e r o f e r r o r s i s B i n o m i a l, W i t h p a r a m e t e r s n a n d p . W e h a v e T h e e x p e c t e d n u m b e r, n p = 0.2 B u t n i s u n k n o w n . T h u s, p c a n n o t b e d e t e r m i n e d . T h e n P o i s s o n a p p r o x i m a t i o n c a n b e u s e d . W i t h p a r a m e t e r λ = n p = 0.2 L e t Y b e t h e n u m b e r o f e r r o r s h a v i n g a p p r o x . P o i s (0.2) d i s t r i b u t i o n . P (Y = k) = \frac{λ^{k} e^{- λ}}{k!} T h u s, P (Y = 0) = \frac{0 . 2^{0} e^{- 0.2}}{0!} = e^{- 0.2} = 0.82$

Given Information

$E x p e c t e d n u m b e r o f t y p o g r a p h i c a l e r r o r s i s 0.2 . S u c h t h a t n p = 0.2 T h e n u m b e r o f l e t t e r i s a s s u m e d t o b e v e r y, v e r y h i g h . S i n c e t h e d i s t r i b u t i o n o f n u m b e r o f e r r o r s i s B i n o m i a l, W i t h p a r a m e t e r s n a n d p . W e h a v e T h e e x p e c t e d n u m b e r, n p = 0.2 B u t n i s u n k n o w n . T h u s, p c a n n o t b e d e t e r m i n e d . T h e n P o i s s o n a p p r o x i m a t i o n c a n b e u s e d . W i t h p a r a m e t e r λ = n p = 0.2 L e t Y b e t h e n u m b e r o f e r r o r s h a v i n g a p p r o x . P o i s (0.2) d i s t r i b u t i o n . P (Y = k) = \frac{λ^{k} e^{- λ}}{k!} T h u s, P (Y \ g e q 2) & = 1 - P (Y = 0) - P (Y = 1) = 1 - e^{- λ} - λ e^{- λ} = 1 - e^{- 0.2} - 0.2 e^{- 0.2} = 1 - 0.82 - 0.164 = 0.016$