{"id":3862,"date":"2026-06-24T11:08:47","date_gmt":"2026-06-24T10:08:47","guid":{"rendered":"https:\/\/www.gironi.it\/blog\/probability-distributions\/"},"modified":"2026-06-28T09:54:00","modified_gmt":"2026-06-28T08:54:00","slug":"probability-distributions","status":"publish","type":"page","link":"https:\/\/www.gironi.it\/blog\/en\/probability-distributions\/","title":{"rendered":"Probability Distributions: an Ordered Guide, from Discrete to Continuous"},"content":{"rendered":"\n

Every time we reason about the uncertain \u2014 how many conversions a campaign will bring, how many visitors will arrive tomorrow, how long a session will last \u2014 we are, without noticing, imagining a shape. Not a single number, but the range of possible values and the weight each of them carries. That shape has a name: it is called a probability distribution, and it is the way statistics describes the behaviour of a quantity we cannot predict with certainty. A probability distribution is nothing more than the map of everything that could happen, with, next to each outcome, how plausible it is that it actually happens.<\/strong> Without this map, every probability calculation remains an exercise disconnected from the reality we would like to measure.<\/p>\n\n\n\n

Knowing the distributions, however, does not mean memorising a catalogue of curves. It means understanding that behind seemingly different problems \u2014 the number of clicks on an ad, the wait before the first sale, the spread of a set of loading times \u2014 hide a few recurring patterns, and that recognising the right pattern is half the work. Once the correct distribution is framed, the practical questions (“how probable is it to exceed this threshold?”, “what value is it reasonable to expect?”) find a solid answer instead of a guess.<\/p>\n\n\n\n

This page is the map of those maps, in order. We do not re-explain the theory here: each stage is an article on the blog, and the order in which we have arranged them is the order in which it makes sense to read them. Anyone starting from scratch can follow them in sequence, from first to last; anyone with some grounding can jump to the group they need. The great division to keep in mind is a single one, and it runs through everything: there are quantities that we count \u2014 events, successes, occurrences, whole and separate numbers \u2014 and quantities that we measure \u2014 times, lengths, proportions, values that slide without jumps along a continuous scale. The former live in discrete distributions<\/em>, the latter in continuous distributions<\/em>. We start with the ones we count.<\/p>\n\n\n\n

Discrete distributions<\/h2>\n\n\n\n

A distribution is discrete when the possible values are separate and countable: zero conversions, one, two, three, never two and a half. It is the world of counts, and it is also the most intuitive, because it coincides with the way we count things in everyday life.
When the question begins with “how many” or “how many times”, the answer almost always lives in a discrete distribution.<\/strong><\/p>\n\n\n\n

The binomial distribution<\/a> is the non-negotiable starting point, and the most intuitive of the group. It describes how many times a certain outcome occurs \u2014 a click, a conversion, a yes \u2014 in a fixed number of independent trials, each with the same probability of success. It is the pattern that returns every time we repeat the same trial several times and count the successes, and it is the conceptual basis on which nearly all the other discrete distributions rest.<\/p>\n\n\n\n

The geometric distribution<\/a> changes the question: no longer how many<\/em> successes in a fixed number of trials, but how many trials<\/em> are needed before the first success appears. It is the distribution of waiting \u2014 how many visitors arrive before the first conversion, how many emails before the first reply \u2014 and grasping it teaches us to reason about the time that precedes an event, not just about its frequency.<\/p>\n\n\n\n

The hypergeometric distribution<\/a> comes into play when the trials are not independent because we draw without replacement<\/em>. Picking customers from a finite list, sampling products from a batch without putting them back: each draw changes the probabilities of the following ones. It is the binomial’s cousin for small populations, and confusing the two is a subtle but frequent mistake.<\/p>\n\n\n\n

The negative binomial distribution<\/a>, also known as the Pascal<\/em> distribution, generalises the geometric one: no longer the wait for the first success, but for the k<\/em>-th. How many trials are needed to gather, say, ten conversions? It is the right tool when the goal is not a single event but a cumulative target, and it shows how a few basic ideas combine into ever richer models.<\/p>\n\n\n\n

The Poisson distribution<\/a> closes the section by moving to counts in time or space: how many rare events happen in a given interval \u2014 visits per hour, errors per day, requests per minute \u2014 when each is independent of the others. It is among the most used distributions in traffic and log analysis, and it marks the natural bridge towards the continuous world, because it is born precisely as a limit of the binomial.<\/p>\n\n\n\n

Continuous distributions<\/h2>\n\n\n\n

A distribution is continuous when the quantity can take any value on a scale without jumps: a time, a length, a proportion. Here it no longer makes sense to ask the probability of an exact value \u2014 it is always zero \u2014 but rather the probability of falling within an interval, and the shape of the curve becomes the real protagonist.
When the question begins with “how much” while measuring, and not with “how many” while counting, we have crossed into the territory of continuous distributions.<\/strong><\/p>\n\n\n\n

The normal distribution<\/a> is the undisputed queen, the bell curve that reappears everywhere. It describes quantities that cluster symmetrically around a central value, with the extremes ever rarer, and it is the other natural candidate as the entry point to the whole path. Its omnipresence is not an accident but a deep consequence of how many small random variations add up, and for this very reason it is the distribution on which the inferential statistics we will meet later rests.<\/p>\n\n\n\n

The Beta distribution<\/a> is less famous but extremely valuable when what we measure is a proportion, a value forced to live between zero and one: a conversion rate, a probability we want to estimate, a percentage. It is flexible as few are \u2014 it can take very different shapes depending on the parameters \u2014 and it is the tool of choice when we want to reason about the uncertainty around a percentage rather than about a single value of it.<\/p>\n\n\n\n

The t-distribution<\/a> closes the path by acting as a hinge towards inference. It resembles the normal one but is a little more cautious, with thicker tails, because it accounts for the fact that, with small samples, we do not know the population’s true variability. It is the distribution we meet as soon as we begin to test<\/em> hypotheses on the data, and for this reason it is the natural passage from the catalogue of distributions to the statistics that uses them to decide.<\/p>\n\n\n\n

Where to start<\/h2>\n\n\n\n

If this is the first contact with the subject, the entry points are two, depending on what we have in front of us. If we are reasoning about counts and successes, the binomial distribution<\/a> is the most intuitive stage to start from: once it is grasped, the geometric, Pascal and Poisson become its natural variations. If instead we are measuring continuous quantities, the door is the normal distribution<\/a>, the curve from which all the rest of the continuous world takes its bearings. Tackle the others out of sequence and, sooner or later, we always come back to these two.<\/p>\n\n\n\n

This is one of the thematic paths we are building to navigate the blog’s articles: the distributions are the vocabulary with which probability describes the uncertain, and they come in handy in every later analysis. The natural next step, once these curves are familiar, is the path devoted to inferential statistics<\/a>: it is there that the right distribution becomes the tool for deciding whether an effect is real or only apparent.<\/p>\n","protected":false},"excerpt":{"rendered":"

Every time we reason about the uncertain \u2014 how many conversions a campaign will bring, how many visitors will arrive tomorrow, how long a session will last \u2014 we are, without noticing, imagining a shape. Not a single number, but the range of possible values and the weight each of them carries. That shape has … Leggi tutto “Probability Distributions: an Ordered Guide, from Discrete to Continuous”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_uag_custom_page_level_css":"","footnotes":""},"class_list":["post-3862","page","type-page","status-publish","hentry"],"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"post-thumbnail":false},"uagb_author_info":{"display_name":"Paolo Gironi","author_link":"https:\/\/www.gironi.it\/blog\/author\/autore-articoli\/"},"uagb_comment_info":0,"uagb_excerpt":"Every time we reason about the uncertain \u2014 how many conversions a campaign will bring, how many visitors will arrive tomorrow, how long a session will last \u2014 we are, without noticing, imagining a shape. Not a single number, but the range of possible values and the weight each of them carries. That shape has…","_links":{"self":[{"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/pages\/3862","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/comments?post=3862"}],"version-history":[{"count":1,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/pages\/3862\/revisions"}],"predecessor-version":[{"id":3907,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/pages\/3862\/revisions\/3907"}],"wp:attachment":[{"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/media?parent=3862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}