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Question Number 102089 by Ar Brandon last updated on 06/Jul/20 $$ \\ $$$$\mathrm{The}\:\mathrm{Gamma}\:\mathrm{function}\:\Gamma\left(\alpha\right)\:\mathrm{is}\:\mathrm{defined}\:\mathrm{as}\:\mathrm{follows}; \\ $$$$\Gamma\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \mathrm{y}^{\alpha−\mathrm{1}} \mathrm{e}^{−\mathrm{y}} \mathrm{dy}\:,\:\alpha>\mathrm{0} \\ $$$$\mathrm{a}\backslash\:\mathrm{Show}\:\mathrm{that}\:\Gamma\left(\alpha+\mathrm{1}\right)=\alpha\Gamma\left(\alpha\right). \\ $$$$\mathrm{b}\backslash\mathrm{Conclude}\:\mathrm{that}\:\Gamma\left(\mathrm{n}\right)=\left(\mathrm{n}−\mathrm{1}\right)!\:,\:\mathrm{n}=\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:… \\ $$$$\mathrm{c}\backslash\mathrm{Determine}\:\Gamma\left(\mathrm{55}\right).…
Question Number 102080 by Ar Brandon last updated on 06/Jul/20 $$\mathrm{A}\:\mathrm{random}\:\mathrm{variable},\:\mathrm{X},\:\mathrm{has}\:\mathrm{a}\:\mathrm{Gamma}\:\mathrm{distribution}\:\mathrm{with} \\ $$$$\mathrm{parameters}\:\alpha\:\mathrm{and}\:\beta,\:\left(\alpha,\:\beta>\mathrm{0}\right).\:\mathrm{The}\:\mathrm{p}.\mathrm{d}.\mathrm{f}\:\mathrm{has}\:\mathrm{the}\:\mathrm{form} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\Gamma\left(\alpha\right)\beta^{\alpha} }\mathrm{x}^{\mathrm{n}−\mathrm{1}} \mathrm{e}^{−\mathrm{x}/\beta} ,\:\mathrm{for}\:\mathrm{x}>\mathrm{0}\:\:,\:\:\Gamma\left(\alpha\right)=\frac{\mathrm{1}}{\beta^{\alpha} }\int_{\mathrm{0}} ^{\infty} \mathrm{x}^{\alpha−\mathrm{1}} \mathrm{e}^{−\mathrm{x}} \mathrm{dx} \\ $$$$\mathrm{a}\backslash\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{Gamma}\:\mathrm{density}\:\mathrm{is}\:\mathrm{a}\:\mathrm{proper}\:\mathrm{p}.\mathrm{d}.\mathrm{f}.…
Question Number 102076 by pete last updated on 06/Jul/20 $$\mathrm{A}\:\mathrm{car}\:\mathrm{is}\:\mathrm{currently}\:\mathrm{valued}\:\mathrm{at}\:\$\mathrm{70350}.\mathrm{00}. \\ $$$$\mathrm{If}\:\mathrm{it}\:\mathrm{loses}\:\mathrm{12\%}\:\mathrm{of}\:\mathrm{its}\:\mathrm{value}\:\mathrm{at}\:\mathrm{the}\:\mathrm{beginning} \\ $$$$\mathrm{of}\:\mathrm{each}\:\mathrm{year}, \\ $$$$\left.\mathrm{a}\right)\:\mathrm{find}\:\mathrm{its}\:\mathrm{value}\:\mathrm{after}\:\mathrm{three}\:\mathrm{and}\:\mathrm{half}\:\mathrm{years}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{depreciation}\:\mathrm{after}\:\mathrm{three}\:\mathrm{years} \\ $$ Commented by PRITHWISH SEN 2…
Question Number 36475 by Rio Mike last updated on 02/Jun/18 $$\:\mathrm{2}\:{regtangle}\:{A}\:{and}\:{B}\:{given}\:{that} \\ $$$${the}\:{legnth}\:{of}\:{A}\:{is}\:{three}\:{times}\:{the} \\ $$$${legnth}\:{of}\:{B},{and}\:{the}\:{Perimeter}\:{of} \\ $$$${A}\:{is}\:\mathrm{102}{cm}.\:{If}\:{the}\:{Area}\:{of}\:{B}\:{is}\: \\ $$$${same}\:{as}\:\mathrm{2}\:{times}\:{the}\:{Area}\:{of}\:{a}\:{Trapezium} \\ $$$${with}\:\:\mathrm{2}\:{parrallel}\:{sides}\:\mathrm{5}{cm}\:{and}\:\mathrm{4}{cm} \\ $$$${with}\:{hieght}\:\mathrm{6}{cm}.\:{find}\:{the}\:{lengths} \\ $$$${and}\:{widths}\:{of}\:{A}\:{and}\:{B}.…
Question Number 36472 by Rio Mike last updated on 02/Jun/18 $${what}\:{is}\:{the}\:{area}\:{of}\:{the}\:{sector}\:{of}\:{a}\:{circle} \\ $$$${with}\:{Radius}\:\mathrm{3}\frac{\mathrm{1}}{\mathrm{2}}{cm}\:{at}\:\mathrm{30}°\:{from}\:{the} \\ $$$${centre}\:\left({take}\:\pi=\:\frac{\mathrm{22}}{\mathrm{7}}\right). \\ $$ Answered by math1967 last updated on 02/Jun/18 $$\frac{\mathrm{30}°}{\mathrm{360}°}×\pi×\left(\frac{\mathrm{7}}{\mathrm{2}}\right)^{\mathrm{2}}…
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Question Number 167468 by LEKOUMA last updated on 17/Mar/22 Answered by MJS_new last updated on 17/Mar/22 $$\mathrm{ln}\:{n}!\:\approx\mathrm{ln}\:\left(\left(\frac{{n}}{\mathrm{e}}\right)^{{n}} \sqrt{\mathrm{2}\pi{n}}\right)\:=\frac{\mathrm{ln}\:\mathrm{2}\pi}{\mathrm{2}}−{n}+\left({n}+\mathrm{1}/\mathrm{2}\right)\mathrm{ln}\:{n} \\ $$$$\mathrm{ln}\:{n}^{{n}} \:={n}\mathrm{ln}\:{n} \\ $$$$\frac{\mathrm{ln}\:{n}!}{\mathrm{ln}\:{n}^{{n}} }=\mathrm{1}+\frac{\mathrm{ln}\:\mathrm{2}\pi}{\mathrm{2}{n}\mathrm{ln}\:{n}}+\frac{\mathrm{1}}{\mathrm{2}{n}}−\frac{\mathrm{1}}{\mathrm{ln}\:{n}} \\…
Question Number 36389 by Rio Mike last updated on 01/Jun/18 $$\:\mathrm{If}\:\mu\:\mathrm{is}\:\mathrm{a}\:\mathrm{universal}\:\mathrm{set}\:\mathrm{and}\:\mathrm{A},\mathrm{B},\mathrm{C}\:\mathrm{are} \\ $$$$\mathrm{3}\:\mathrm{sets}\:.\:\:\:\mathrm{Given}\:\mathrm{that}\: \\ $$$$\mathrm{n}\left(\mu\right)=\:\mathrm{52}\:\:,\:\mathrm{n}\left(\mathrm{A}\cap\mathrm{B}\right)\:=\:\mathrm{26}\:\mathrm{and}\: \\ $$$$\mathrm{n}\left(\mathrm{A}\:\cup\:\mathrm{C}\right)\:=\:\mathrm{32}\:\mathrm{find}\:\mathrm{n}\left(\mathrm{A}\right),\mathrm{n}\left(\mathrm{B}\right)\:\mathrm{and} \\ $$$$\mathrm{n}\left(\mathrm{C}\right) \\ $$ Terms of Service Privacy…
Question Number 36385 by solihin last updated on 01/Jun/18 Commented by solihin last updated on 01/Jun/18 $$\mathrm{10}.{b}? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…