{"id":7609,"date":"2022-04-15T17:47:46","date_gmt":"2022-04-15T12:17:46","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7609"},"modified":"2025-06-02T20:50:51","modified_gmt":"2025-06-02T15:20:51","slug":"what-is-exponential-distribution%ef%bf%bc","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-exponential-distribution%ef%bf%bc\/","title":{"rendered":"What is exponential distribution?\ufffc"},"content":{"rendered":"\n<p>The<a href=\"https:\/\/www.itl.nist.gov\/div898\/handbook\/eda\/section3\/eda3667.htm\" target=\"_blank\" rel=\"noreferrer noopener\"> exponential distribution<\/a> is a continuous probability distribution and we widely use this distribution in various fields such as engineering, research, data sciences etc.<\/p>\n\n\n\n<p>We use this to model the time elapsed between two events in a Poisson process.<\/p>\n\n\n\n<p>There is a very strong relationship between Poisson and exponential distribution.<\/p>\n\n\n\n<p>For example, suppose a Poisson distribution models the number of cars passes through the highway in a given time period. We can model the time in between each car with an exponential distribution.<\/p>\n\n\n\n<p>[For Poisson distribution- Read our article]<\/p>\n\n\n\n<p>Here we will mathematically define the exponential distribution.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Definition <\/h2>\n\n\n\n<p>Many problems involve the measurement of the duration of time X between an initial point of time and the occurrence of some phenomenon of interest. For example X is the life time of a light bulb, you turn it on and left until it burns out. The continuous random variable X having the probability density function<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/KLf0wABnMEfnYCk5rbXF_kOKRefC1tnISay65Q7F4QzNRzGy4Se0d56Ghvad3cSu7lTCeTMzPgEjnvuJNiZtVgcyS7-Y8jjCMtqW0m6cSFKcr-xJsao5a7B0Iq3k005k6oaFgJQc\" alt=\"\" width=\"201\" height=\"49\" \/><\/figure>\n\n\n\n<p>Where, the rate \u03bb is the Normal amount of events in single time.<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/qIEJafVqiegysTI5mywea7UvK8QuviuBjRgkd-wKvfg4_lK1rc2S3R796k2UnnHDySJQkWMhJftiWeqD0PxWflcKsqFMxPaey-MyJhzCS_M260lbqrW6Ank55S_7QJ7TCXf87Opb\" alt=\"\" width=\"247\" height=\"223\" \/><\/figure>\n\n\n\n<p>Here the only parameter of the distribution is \u03bb which is greater than zero. This distribution, also known as the negative exponential distribution, is a special case of the gamma distribution (with r = 1). Examples of random variables modeled as exponential are-<\/p>\n\n\n\n<p>a. (inter-arrival) time between two successive job arrivals<\/p>\n\n\n\n<p>b. duration of telephone calls<\/p>\n\n\n\n<p>c. life time (or time to failure) of a component or a product<\/p>\n\n\n\n<p>d. service time at a server in a queue<\/p>\n\n\n\n<p>e. time required for repair of a component<\/p>\n\n\n\n<p>The mean value is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/LjgtxMPGB35T-Hmv1aOOqw2J1T06g1EnbnJJ_727-hmHC5D3en1WfEcUhONKJ8sF2xM5AAxVHjPZ6-ZyHjdPPlm-8xzM62gNpmmG3kmP9hXRSslqBd2Jbf_OTVFhOTaqOHsoMJVT\" width=\"46\" height=\"37\">. The median of the exponential distribution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/cBTg81Dz-JYCjFWyDJK9Vgsj2IPz7nf2M22walszK2Ui3_nh8YrFv6P2b0RctQqNisSWtGtLTcg_1COOgaAXFyIpWoZF-JIg2FUZMMUp-iaIjoiYqRrFhqvccQvvIOSRmHV2Gxva\" width=\"56\" height=\"33\">, and the variance is shown by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/fajriEsofWhhN-ybJSjayjAAEEOAl5O94KfetULkRFlVyIN06uTI-hiC2St1vQ_GvRz55f0xZQRiRAZCEpR6xx5d39MBc6ZtQinGT9MEDD0v2OzcP9AjMGz7q2oX9ONsfHvVwN8b\" width=\"60\" height=\"33\"><\/p>\n\n\n\n<p>Note: Both the mean and standard deviation of the exponential distribution are equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/v7dGct-cpSlF4l76Mf6z58fOxb9vvVO6pUQAivLComoRs8fdxajYB4M23srEHkHgcWiuQ0MFckYJoZkq9c2JRlFpeNeZDxwAvAkaK_B-MeaeDhqiGkVy-GTMqNHoSJ9UZm2ZwXhR\" width=\"23\" height=\"32\"><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Example<\/strong><\/h2>\n\n\n\n<p><strong>The length of time for one person to be served at a restaurant is a random variable X having an exponential distribution with a mean of 4 minutes. Find the probability that <\/strong>the <strong>restaurant s<\/strong>erves a <strong>person<\/strong> <strong>in less than 3 minutes on at least 4 of the next 6 days.<\/strong><\/p>\n\n\n\n<p>Sol:<\/p>\n\n\n\n<p>The probability that a a restaurant serves a person at a cafeteria in less than 3 minutes is-&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/PoAqjfIzGtYzW7oZaZ5T14IF2UtkrAc9DwLgO43mf6Lrgghz_NDCbuKllSAVwEawsmZKxBvDveSBu47ugKzk2YMvMUC65uZSELAnNkcuRV09MGP4cUWEENS-TIEOihyXxY9FT3Wj\" alt=\"\" \/><\/figure>\n\n\n\n<p>Since the mean \u03bc = 1\/ \u03bb = 4 or \u03bb = 1\/4 , the exponential distribution is- <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/H-8mg8yHUmDL6k9Ly0pKRFz3fFU7p8kCNxwg3ncZ4HUE7p885twlPOJi3YBrBsuap9nQEq7dcDNidN689hi0_dOcZ0HyBK3aDE7Q02JW0tbDusWjLkIVPr5KPbIHQb_rjCkRSxZs\" width=\"44\" height=\"39\"><\/p>\n\n\n\n<p>Now<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/1i3izPTfwiSZyvcHKa_EKPT4Hnp7PcajODq2uC7ZFGx01Jccvhyks8mnuTQcamyfVOnuevnyikAyDKWshNu8M10a_P3qb8PgsUKgE2q9DVFlj9cwEuxXzk_AG7t83R3AFl43Fopi\" alt=\"\" \/><\/figure>\n\n\n\n<p>Let X represent the number of days on which a person is served in less than 3 minutes. <\/p>\n\n\n\n<p>Then using the binomial distribution, the probability that a person is served in less than 3 minutes on at least 4 of the next 6 days is<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/28VnlI80lb3Ql341G0gWA9UFJt5iAEYN_eHRcLS9j35_Z_JQhgVs7GT9xegXGY2fdCqDoJYGx8wrqwkZBwRVkRil77XoEFWVH4Wh_bd6tfh3oxEtQAdJC3cVpPKzPYSTwgaiwopE\" alt=\"\" \/><\/figure>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The exponential distribution is a continuous probability distribution and it is a widely used distribution in various fields such as engineering, research, data sciences etc.<\/p>\n","protected":false},"author":3,"featured_media":4888,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-7609","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is exponential distribution?\ufffc - Goseeko blog<\/title>\n<meta name=\"description\" content=\"The exponential distribution is a continuous probability distribution and it is a widely used distribution in various fields such as engineering, research, data sciences etc.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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