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Category: Probability and Statistics

Given-a-random-variable-X-of-image-set-X-1-1-2-with-probabilities-P-X-1-e-a-P-X-1-e-b-and-P-X-2-e-c-where-a-b-and-c-are-in-an-Arithmetic-Progression-Assuming-the-mathematical-expec

Question Number 87643 by Ar Brandon last updated on 05/Apr/20 $${Given}\:{a}\:{random}\:{variable}\:\boldsymbol{{X}}\:{of}\:{image}\:{set} \\ $$$${X}\left(\Omega\right)=\left[\mathrm{1};−\mathrm{1};\mathrm{2}\right]\:{with}\:{probabilities}\:{P}\left({X}=\mathrm{1}\right)={e}^{{a}} , \\ $$$${P}\left({X}=−\mathrm{1}\right)={e}^{{b}} ,\:{and}\:{P}\left({X}=\mathrm{2}\right)={e}^{{c}} \:{where}\:{a},\:{b},\:{and}\:{c}\: \\ $$$${are}\:{in}\:\:{an}\:{Arithmetic}\:{Progression}. \\ $$$${Assuming}\:{the}\:{mathematical}\:{expection}\:{E}\left({X}\right)\:{of}\:{X}\: \\ $$$${is}\:{equal}\:{to}\:\mathrm{1}. \\…

In-bottle-manufacturing-company-it-was-observed-that-5-of-the-bottles-manufactured-were-defective-In-a-random-sample-of-150-bottles-find-probability-that-a-exactly-3-b-between-3-and-6-

Question Number 152940 by nadovic last updated on 03/Sep/21 $$\:\mathrm{In}\:\mathrm{bottle}\:\mathrm{manufacturing}\:\mathrm{company},\:\mathrm{it} \\ $$$$\mathrm{was}\:\mathrm{observed}\:\mathrm{that}\:\mathrm{5\%}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bottles} \\ $$$$\mathrm{manufactured}\:\mathrm{were}\:\mathrm{defective}.\:\mathrm{In}\:\mathrm{a}\: \\ $$$$\mathrm{random}\:\mathrm{sample}\:\mathrm{of}\:\mathrm{150}\:\mathrm{bottles},\:\mathrm{find}\: \\ $$$$\mathrm{probability}\:\mathrm{that}\: \\ $$$$\:\left({a}\right)\:\mathrm{exactly}\:\mathrm{3}, \\ $$$$\:\left({b}\right)\:\mathrm{between}\:\mathrm{3}\:\mathrm{and}\:\mathrm{6}, \\ $$$$\:\left({c}\right)\:\mathrm{at}\:\mathrm{most}\:\mathrm{4}, \\…

Question-152937

Question Number 152937 by DELETED last updated on 03/Sep/21 Answered by DELETED last updated on 03/Sep/21 $$\left.\mathrm{4}\right).\:\mathrm{mean}\rightarrow\overset{−} {\mathrm{x}}\:=\frac{\Sigma\mathrm{x}_{\mathrm{i}} ×\mathrm{f}_{\mathrm{i}} }{\Sigma\mathrm{f}_{\mathrm{i}} } \\ $$$$\:\:\:\:\:\overset{−} {\mathrm{x}}=\frac{\mathrm{4}×\mathrm{33}+\mathrm{7}×\mathrm{38}+\mathrm{9}×\mathrm{43}+\mathrm{6}×\mathrm{48}+\mathrm{4}×\mathrm{53}}{\mathrm{30}} \\…

Question-152935

Question Number 152935 by DELETED last updated on 03/Sep/21 Answered by DELETED last updated on 03/Sep/21 $$\left.\mathrm{3}.\mathrm{a}\right).\:\mathrm{Q}_{\mathrm{2}} =\mathrm{L}_{\mathrm{i}} +\left(\frac{\mathrm{N}/\mathrm{2}−<\Sigma\mathrm{f}}{\Sigma\mathrm{f}_{\mathrm{i}} }\right)×\mathrm{C} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left(\mathrm{45}−\mathrm{0}.\mathrm{5}\right)+\left(\frac{\mathrm{40}/\mathrm{2}−\mathrm{16}}{\mathrm{12}}\right)×\mathrm{5} \\ $$$$\:\:\:\:\:=\mathrm{44},\mathrm{5}+\frac{\mathrm{4}×\mathrm{5}}{\mathrm{12}}=\mathrm{44},\mathrm{5}+\mathrm{1},\mathrm{67} \\…

Question-152903

Question Number 152903 by DELETED last updated on 03/Sep/21 Answered by DELETED last updated on 03/Sep/21 $$\left.\mathrm{1}.\mathrm{a}\right).\:\overset{−} {\mathrm{x}}\:=\:\frac{\Sigma\mathrm{f}_{\mathrm{i}} .\mathrm{x}_{\mathrm{i}} }{\Sigma\mathrm{f}_{\mathrm{i}} }\: \\ $$$$\:=\frac{\mathrm{2}×\mathrm{14},\mathrm{5}+\mathrm{9}×\mathrm{18},\mathrm{5}+\mathrm{12}×\mathrm{22},\mathrm{5}+\mathrm{9}×\mathrm{26},\mathrm{5}+\mathrm{5}×\mathrm{30},\mathrm{5}+\mathrm{3}×\mathrm{34},\mathrm{5}}{\mathrm{2}+\mathrm{9}+\mathrm{12}+\mathrm{9}+\mathrm{5}+\mathrm{3}} \\ $$$$=\frac{\mathrm{29}+\mathrm{166},\mathrm{5}+\mathrm{270}+\mathrm{238},\mathrm{5}+\mathrm{152},\mathrm{5}+\mathrm{103},\mathrm{5}}{\mathrm{40}}…

The-probability-that-athlete-will-win-a-race-is-1-6-and-that-he-will-be-second-and-third-are-1-4-and-1-3-respectively-what-is-the-probability-that-he-will-not-be-first-in-the-first-three-place-

Question Number 152683 by rexford last updated on 31/Aug/21 $${The}\:{probability}\:{that}\:{athlete}\:{will}\:{win}\:{a}\:{race}\:{is}\:\frac{\mathrm{1}}{\mathrm{6}}\:{and}\:{that} \\ $$$${he}\:{will}\:{be}\:{second}\:{and}\:{third}\:{are}\:\frac{\mathrm{1}}{\mathrm{4}}\:{and}\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$${respectively}.{what}\:{is}\:{the}\:{probability}\:{that}\:{he}\:{will}\:{not}\:{be}\:{first} \\ $$$${in}\:{the}\:{first}\:{three}\:{place}! \\ $$$${Please},{help}\:{me}\:{out} \\ $$ Answered by Olaf_Thorendsen last updated…

what-is-P-x-gt-1-if-x-has-a-PDF-of-f-x-1-4-2-lt-x-lt-2-0-elsewhere-

Question Number 86668 by jagoll last updated on 30/Mar/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{P}\left(\mid\mathrm{x}\mid\:>\:\mathrm{1}\:\right)\:\mathrm{if}\:\mathrm{x}\:\mathrm{has}\:\mathrm{a}\:\mathrm{PDF}\:\mathrm{of} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\begin{cases}{\frac{\mathrm{1}}{\mathrm{4}}\:,\:\:−\mathrm{2}<\mathrm{x}<\mathrm{2}}\\{\mathrm{0}\:,\:\mathrm{elsewhere}}\end{cases} \\ $$ Commented by john santu last updated on 30/Mar/20 $$\Rightarrow\mathrm{P}\left(\mathrm{x}<−\mathrm{1}\:\cup\:\mathrm{x}>\mathrm{1}\right) \\ $$$$=\:\int_{−\mathrm{2}}…