{"id":65526,"date":"2026-09-29T14:37:17","date_gmt":"2026-09-29T11:37:17","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65526"},"modified":"2026-09-29T15:19:13","modified_gmt":"2026-09-29T12:19:13","slug":"exponential-distribution-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/exponential-distribution-calculator\/","title":{"rendered":"Exponential Distribution Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Compute Exponential Distribution Probabilities<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65526\" data-favorite-title=\"Exponential Distribution\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/exponential-distribution-calculator\/\" data-favorite-excerpt=\"Calculate exponential distribution probabilities from a rate \u03bb or a mean waiting time: density, P(X \u2264 t), P(X &gt; t), interval probability, mean and median. Enter the rate, a time, and an optional interval end.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-wpcalc-distribution-histogram\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/statistics\/exponential-distribution-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65526\" data-post=\"65526\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Specify parameter by<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"specify_parameter_by\"><option value=\"rate\" selected>Rate \u03bb (events per time)<\/option><option value=\"mean\">Mean waiting time \u03bc<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Rate (\u03bb)<\/label><div class=\"term\">\u03bb<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"rate_\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"2\"\/><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"per\" aria-label=\"per\"><option value=\"1\">per second<\/option><option value=\"60\">per minute<\/option><option value=\"3600\" selected>per hour<\/option><option value=\"86400\">per day<\/option><option value=\"31536000\">per year<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-2\">Average number of events per unit of time.<\/small><\/div><div class=\"field has-term\" id=\"input-3\" hidden><label for=\"field-4\">Mean waiting time (\u03bc)<\/label><div class=\"term\">\u03bc<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"mean_waiting_time_\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"30\"\/><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"in\" aria-label=\"in\"><option value=\"1\">seconds<\/option><option value=\"60\" selected>minutes<\/option><option value=\"3600\">hours<\/option><option value=\"86400\">days<\/option><option value=\"31536000\">years<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-4\">Average time between consecutive events (\u03bb = 1\/\u03bc).<\/small><\/div><div class=\"is-single field\" id=\"input-4\"><span class=\"font-semibold\">Time<\/span><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-7\">Time (t)<\/label><div class=\"term\">t<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"time_t\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"1\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"in\" aria-label=\"in\"><option value=\"1\">seconds<\/option><option value=\"60\">minutes<\/option><option value=\"3600\" selected>hours<\/option><option value=\"86400\">days<\/option><option value=\"31536000\">years<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-7\">The waiting time you are interested in.<\/small><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-9\">Interval end (b)<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"interval_end_b\" id=\"field-9\" aria-describedby=\"calc-info-9\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"2\"\/><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"in\" aria-label=\"in\"><option value=\"1\">seconds<\/option><option value=\"60\">minutes<\/option><option value=\"3600\" selected>hours<\/option><option value=\"86400\">days<\/option><option value=\"31536000\">years<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-9\">Optional: fill in to also get P(t \u2264 X \u2264 b). Leave empty to skip.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65526\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Exponential Distribution Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Density f(t)<\/span><span class=\"result-term\">f(t)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Cumulative probability P(X \u2264 t)<\/span><span class=\"result-term\">F(t)<\/span><\/span><span class=\"field-value\" id=\"result-2\" data-unit=\"%\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Survival probability P(X &gt; t)<\/span><span class=\"result-term\">S(t)<\/span><\/span><span class=\"field-value\" id=\"result-3\" data-unit=\"%\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Interval probability P(t \u2264 X \u2264 b)<\/span><\/span><span class=\"field-value\" id=\"result-4\" data-unit=\"%\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Mean waiting time E(X)<\/span><span class=\"result-term\">E(X)<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-6\"><span class=\"field-title\"><span>Formula, median &amp; interpretation<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"65526\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Exponential Distribution Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ f(t) = \\lambda e^{-\\lambda t}, \\qquad F(t) = P(X \\le t) = 1 - e^{-\\lambda t} \\]\n\\[ S(t) = P(X &gt; t) = e^{-\\lambda t}, \\qquad P(a \\le X \\le b) = e^{-\\lambda a} - e^{-\\lambda b} \\]\n\\[ E(X) = \\frac{1}{\\lambda}, \\qquad \\text{median} = \\frac{\\ln 2}{\\lambda} \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(\\lambda\\) \u2014 rate parameter: the average number of events per unit of time<\/li>\n<li>\\(t\\) \u2014 the waiting time of interest<\/li>\n<li>\\(f(t)\\) \u2014 probability density at time \\(t\\)<\/li>\n<li>\\(F(t)\\) \u2014 cumulative probability of waiting no longer than \\(t\\)<\/li>\n<li>\\(S(t)\\) \u2014 survival probability of waiting longer than \\(t\\)<\/li>\n<\/ul>\n<p>The exponential distribution models the time between consecutive events that occur independently at a constant average rate. Its single parameter \\(\\lambda\\) fully determines everything; the mean waiting time is simply \\(1\/\\lambda\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Exponential Distribution \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> calls arrive at a call center at an average rate of 2 per hour. What is the probability that the next call arrives within 1 hour, and what is the chance it arrives between 1 and 2 hours?<\/p>\n<p>\\[ \\lambda t = 2 \\times 1 = 2 \\]<\/p>\n<p>\\[ F(1) = 1 - e^{-2} = 1 - 0.1353 = 0.8647 \\]<\/p>\n<p>\\[ P(1 \\le X \\le 2) = e^{-2} - e^{-4} = 0.1353 - 0.0183 = 0.1170 \\]<\/p>\n<p><strong>Result:<\/strong> P(X \u2264 1 h) = 86.47%, P(X &gt; 1 h) = 13.53%, and P(1 h \u2264 X \u2264 2 h) = 11.70%. The density at t = 1 h is f(1) = 2e\u207b\u00b2 \u2248 0.2707 per hour, the mean wait is 0.5 hour (30 minutes), and the median is ln(2)\/2 \u2248 0.3466 hour (~20.8 minutes). These are exactly the values shown by the calculator with its default settings.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Open <strong>Specify parameter by<\/strong> and choose whether you want to enter the rate \\(\\lambda\\) (events per time) or the mean waiting time \\(\\mu\\). Only the matching field is shown.<\/li>\n<li>If you chose the rate, fill in <strong>Rate (\u03bb)<\/strong> and pick the time unit it is expressed in (per second, per minute, per hour, per day, per year). If you chose the mean, fill in <strong>Mean waiting time (\u03bc)<\/strong> with its unit instead; the calculator converts it via \\(\\lambda = 1\/\\mu\\).<\/li>\n<li>Enter <strong>Time (t)<\/strong> \u2014 the waiting time you are interested in \u2014 and select its unit under \"in\".<\/li>\n<li>Optionally fill in <strong>Interval end (b)<\/strong> to also get the probability \\(P(t \\le X \\le b)\\). Leave it empty to skip this result.<\/li>\n<li>Read the results: density \\(f(t)\\), cumulative probability \\(P(X \\le t)\\), survival probability \\(P(X &gt; t)\\), the interval probability, the mean waiting time, and a summary line with the median, the 90th-percentile wait and a plain-language interpretation.<\/li>\n<\/ol>\n<h2>What Is the Exponential Distribution?<\/h2>\n<p>The exponential distribution describes the waiting time until the next event in a process where events happen independently at a constant average rate \u2014 a <em>Poisson process<\/em>. Typical applications include the time between customer arrivals, the time until a radioactive particle decays, the lifetime of an electronic component, or the time until the next server failure.<\/p>\n<p>The distribution is defined by a single parameter, the rate \\(\\lambda\\). The probability density \\(f(t) = \\lambda e^{-\\lambda t}\\) is largest at \\(t = 0\\) and decays smoothly, meaning short waits are the most likely. The cumulative function \\(F(t) = 1 - e^{-\\lambda t}\\) gives the probability that the wait ends at or before time \\(t\\); its complement, the survival function \\(S(t) = e^{-\\lambda t}\\), gives the probability of still waiting after \\(t\\).<\/p>\n<p>The mean and the standard deviation are both \\(1\/\\lambda\\), but the median \\(\\ln(2)\/\\lambda \\approx 0.693\/\\lambda\\) is smaller. This tells you something practical: more than half of all waits end before the average. A large wait is possible but increasingly rare \u2014 for example, only 5% of waits exceed \\(3\/\\lambda\\).<\/p>\n<p>A defining property of the exponential distribution is <strong>memorylessness<\/strong>: \\(P(X &gt; s + t \\mid X &gt; s) = P(X &gt; t)\\). If a light bulb has an exponentially distributed lifetime and has already burned for a year, its chances of surviving another year are exactly the same as when it was new. This makes the distribution the continuous analogue of the geometric distribution and the natural companion of the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/poisson-distribution-calculator\/\">Poisson distribution<\/a>, which describes the <em>number<\/em> of events in a fixed period rather than the time between them.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Constant rate.<\/strong> The model assumes events occur at a steady average rate at all times. If the rate varies (e.g., rush-hour arrivals, wear-out of mechanical parts), the exponential distribution will under- or overestimate the probabilities.<\/li>\n<li><strong>Independence.<\/strong> One event must not influence the timing of the next. Clustering of events violates this assumption.<\/li>\n<li><strong>Memorylessness.<\/strong> If the process truly is exponential, already-elapsed waiting time carries no information about the remaining wait. Real lifetimes of machines and humans usually are not memoryless; for those, reliability models such as the Weibull distribution fit better.<\/li>\n<li><strong>Units.<\/strong> The calculator converts all units internally, so you can enter the rate per hour and the time in minutes without doing any arithmetic yourself. One year is taken as 365 days.<\/li>\n<li><strong>Density vs. probability.<\/strong> \\(f(t)\\) is a density per time unit, not a probability \u2014 it can exceed 1. The meaningful probability statements are \\(F(t)\\), \\(S(t)\\) and the interval probability.<\/li>\n<li>If you need to model counts of events rather than waiting times, use the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/poisson-distribution-calculator\/\">Poisson distribution calculator<\/a>; for yes\/no outcomes over a fixed number of trials, see the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/binomial-distribution-calculator\/\">binomial distribution calculator<\/a>, and for symmetric continuous data the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/normal-distribution-calculator\/\">normal distribution calculator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculate exponential distribution probabilities from a rate \u03bb or a mean waiting time: density, P(X \u2264 t), P(X > t), interval probability, mean and median. Enter the rate, a time, and an optional interval end.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[74],"tags":[77],"class_list":["post-65526","post","type-post","status-publish","format-standard","hentry","category-statistics","tag-distributions"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65526","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65526"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65526\/revisions"}],"predecessor-version":[{"id":66405,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65526\/revisions\/66405"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65526"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65526"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65526"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}