To solve this problem, we need to model the situation using a Poisson process, which is a mathematical concept used to describe events occurring at a constant average rate over time. In this case, hurricanes make landfall in North Carolina according to a Poisson process with a rate of 1 per 3 years.<\/p>\n\n\n\n
Poisson Process:<\/h3>\n\n\n\n
A Poisson process describes the number of events (hurricanes) that occur in a given time interval, where the events occur independently, and the average number of events is constant over time.<\/p>\n\n\n\n
- \n
- Rate of Occurrence (\u03bb)<\/strong>: The rate of hurricanes making landfall is given as 1 every 3 years. This means that the average number of hurricanes per year is \u03bb=13\\lambda = \\frac{1}{3}\u03bb=31\u200b hurricanes per year.<\/li>\n\n\n\n
- Poisson Distribution<\/strong>: The Poisson distribution is used to calculate the probability of a certain number of events occurring in a fixed time interval. For a Poisson process, the probability of observing kkk events in a time interval of length ttt is given by the formula:<\/li>\n<\/ul>\n\n\n\n
P(k events in time t)=(\u03bbt)ke\u2212\u03bbtk!P(k \\text{ events in time } t) = \\frac{(\\lambda t)^k e^{-\\lambda t}}{k!}P(k events in time t)=k!(\u03bbt)ke\u2212\u03bbt\u200b<\/p>\n\n\n\n
Where:<\/p>\n\n\n\n
- \n
- \u03bb\\lambda\u03bb is the rate of occurrence (1 hurricane per 3 years, or 13\\frac{1}{3}31\u200b per year),<\/li>\n\n\n\n
- ttt is the length of time in years,<\/li>\n\n\n\n
- kkk is the number of hurricanes we are interested in, and<\/li>\n\n\n\n
- eee is Euler\u2019s number (approximately 2.718).<\/li>\n<\/ul>\n\n\n\n
Example Calculation:<\/h3>\n\n\n\n
If we want to find the probability of exactly 2 hurricanes making landfall in North Carolina within a 5-year period, we would calculate it as follows:<\/p>\n\n\n\n
- \n
- Calculate \u03bbt\\lambda t\u03bbt<\/strong>:
\u03bb=13\\lambda = \\frac{1}{3}\u03bb=31\u200b hurricanes per year, and t=5t = 5t=5 years, so: \u03bbt=13\u00d75=53\u22481.67\\lambda t = \\frac{1}{3} \\times 5 = \\frac{5}{3} \\approx 1.67\u03bbt=31\u200b\u00d75=35\u200b\u22481.67<\/li>\n\n\n\n- Apply the Poisson formula<\/strong> for k=2k = 2k=2 hurricanes: P(k=2)=(1.67)2e\u22121.672!P(k = 2) = \\frac{(1.67)^2 e^{-1.67}}{2!}P(k=2)=2!(1.67)2e\u22121.67\u200b After calculating the above expression, you would get the probability of exactly 2 hurricanes making landfall in the next 5 years.<\/li>\n<\/ol>\n\n\n\n
Interpretation:<\/h3>\n\n\n\n
This type of problem is useful for understanding how likely certain events are to happen within a given time frame, assuming a constant average rate. The Poisson process helps model real-world phenomena like hurricanes, phone calls arriving at a call center, or traffic flow at a busy intersection, where the events occur randomly but with a known average rate.<\/p>\n\n\n\n
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<\/figure>\n","protected":false},"excerpt":{"rendered":"Assume hurricanes make landfall in North Carolina according to a Poisson process. They make landfall at a rate of one per three years. The Correct Answer and Explanation is: To solve this problem, we need to model the situation using a Poisson process, which is a mathematical concept used to describe events occurring at a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-261981","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/261981","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=261981"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/261981\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=261981"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=261981"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=261981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Apply the Poisson formula<\/strong> for k=2k = 2k=2 hurricanes: P(k=2)=(1.67)2e\u22121.672!P(k = 2) = \\frac{(1.67)^2 e^{-1.67}}{2!}P(k=2)=2!(1.67)2e\u22121.67\u200b After calculating the above expression, you would get the probability of exactly 2 hurricanes making landfall in the next 5 years.<\/li>\n<\/ol>\n\n\n\n
- Calculate \u03bbt\\lambda t\u03bbt<\/strong>:
- Poisson Distribution<\/strong>: The Poisson distribution is used to calculate the probability of a certain number of events occurring in a fixed time interval. For a Poisson process, the probability of observing kkk events in a time interval of length ttt is given by the formula:<\/li>\n<\/ul>\n\n\n\n