The Poisson distribution gives the probability for the
number of occurrences
for a "rare" event. - refer to screenshot
(a) less than 20 days (i.e., x<><20) round="" to="" four="" decimal="">
(b) more than 50 days
0 be a random variable and let B > 0 be a constant. Then y = ex/B is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities. y y= a b X If a and b are any numbers such that 0 < a="">< b,="" then="" using="" some="" extra="" mathematics,="" it="" can="" be="" shown="" that="" the="" area="" under="" the="" curve="" above="" the="" interval="" [a,="" b]="" is="" the="" following.="" p(a="">< x="">< b)="e-a/ß" -="" e-b/b="" notice="" that="" by="" definition,="" x="" cannot="" be="" negative,="" so,="" p(x="">< 0)="0." the="" random="" variable="" x="" is="" called="" an="" exponential="" random="" variable.="" using="" some="" more="" mathematics,="" it="" can="" be="" shown="" that="" the="" mean="" and="" standard="" deviation="" of="" are="" the="" following.="" µ="B" and="" o="B" note:="" the="" number="" e="2.71828." ..="" is="" used="" throughout="" probability,="" statistics,="" and="" mathematics.="" the="" key="" eš="" is="" conveniently="" located="" on="" most="" calculators.="" comment:="" the="" poisson="" and="" exponential="" distributions="" have="" a="" special="" relationship.="" specifically,="" it="" can="" be="" shown="" that="" the="" waiting="" time="" between="" successive="" poisson="" arrivals="" (i.e.,="" successes="" or="" rare="" events)="" has="" an="" exponential="" distribution="" with="" b="1/2," where="" 1="" is="" the="" average="" number="" of="" poisson="" successes="" (rare="" events)="" per="" unit="" of="" time.="" fatal="" accidents="" on="" scheduled="" domestic="" passenger="" flights="" are="" rare="" events.="" in="" fact,="" airlines="" do="" all="" they="" possibly="" can="" to="" prevent="" such="" accidents.="" however,="" around="" the="" world="" such="" fatal="" accidents="" do="" occur.="" let="" x="" be="" a="" random="" variable="" representing="" the="" waiting="" time="" between="" fatal="" airline="" accidents.="" research="" has="" shown="" that="" x="" has="" an="" exponential="" distribution="" with="" a="" mean="" of="" approximately="" 44="" days.t="" we="" take="" the="" point="" of="" view="" that="" x="" (measured="" in="" days="" as="" units)="" is="" a="" continuous="" random="" variable.="" suppose="" a="" fatal="" airline="" accident="" has="" just="" been="" reported="" on="" the="" news.="" what="" is="" the="" probability="" that="" the="" waiting="" time="" to="" the="" next="" reported="" fatal="" airline="" accident="" is="" the="" following?="" (a)="" less="" than="" 20="" days="" (i.e.,="" 0=""><>< 20)="" (round="" your="" answer="" to="" four="" decimal="" places.)="" "/="">
Extracted text: The Poisson distribution gives the probability for the number of occurrences for a "rare" event. Now, let x be a random variable that represents the waiting time between rare events. Using some mathematics, it can be shown that x has an exponential distribution. Let x > 0 be a random variable and let B > 0 be a constant. Then y = ex/B is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities. y y= a b X If a and b are any numbers such that 0 < a="">< b,="" then="" using="" some="" extra="" mathematics,="" it="" can="" be="" shown="" that="" the="" area="" under="" the="" curve="" above="" the="" interval="" [a,="" b]="" is="" the="" following.="" p(a="">< x="">< b)="e-a/ß" -="" e-b/b="" notice="" that="" by="" definition,="" x="" cannot="" be="" negative,="" so,="" p(x="">< 0)="0." the="" random="" variable="" x="" is="" called="" an="" exponential="" random="" variable.="" using="" some="" more="" mathematics,="" it="" can="" be="" shown="" that="" the="" mean="" and="" standard="" deviation="" of="" are="" the="" following.="" µ="B" and="" o="B" note:="" the="" number="" e="2.71828." ..="" is="" used="" throughout="" probability,="" statistics,="" and="" mathematics.="" the="" key="" eš="" is="" conveniently="" located="" on="" most="" calculators.="" comment:="" the="" poisson="" and="" exponential="" distributions="" have="" a="" special="" relationship.="" specifically,="" it="" can="" be="" shown="" that="" the="" waiting="" time="" between="" successive="" poisson="" arrivals="" (i.e.,="" successes="" or="" rare="" events)="" has="" an="" exponential="" distribution="" with="" b="1/2," where="" 1="" is="" the="" average="" number="" of="" poisson="" successes="" (rare="" events)="" per="" unit="" of="" time.="" fatal="" accidents="" on="" scheduled="" domestic="" passenger="" flights="" are="" rare="" events.="" in="" fact,="" airlines="" do="" all="" they="" possibly="" can="" to="" prevent="" such="" accidents.="" however,="" around="" the="" world="" such="" fatal="" accidents="" do="" occur.="" let="" x="" be="" a="" random="" variable="" representing="" the="" waiting="" time="" between="" fatal="" airline="" accidents.="" research="" has="" shown="" that="" x="" has="" an="" exponential="" distribution="" with="" a="" mean="" of="" approximately="" 44="" days.t="" we="" take="" the="" point="" of="" view="" that="" x="" (measured="" in="" days="" as="" units)="" is="" a="" continuous="" random="" variable.="" suppose="" a="" fatal="" airline="" accident="" has="" just="" been="" reported="" on="" the="" news.="" what="" is="" the="" probability="" that="" the="" waiting="" time="" to="" the="" next="" reported="" fatal="" airline="" accident="" is="" the="" following?="" (a)="" less="" than="" 20="" days="" (i.e.,="" 0=""><>< 20) (round your answer to four decimal places.) 20)="" (round="" your="" answer="" to="" four="" decimal="">