{ "source_file": "0930", "source_platform": "Medium", "author": "Kurtis Pykes", "title_listed": "Probability Distributions To Be Aware Of For Data Science (With Code)", "keywords": "概率分布,统计", "domain": "机器学习", "content_type": "深度技术解析", "is_paid_flag": "部分免费/会员墙", "url": "https://towardsdatascience.com/probability-distributions-to-be-aware-of-for-data-science-with-code-c4a6bb8b0e9a/", "publish_date": "2022-04", "corpus_stage": "预训练/SFT", "notes": "", "id": "2c27fdbe4973", "collected_at": "2026-10-02T00:05:40", "title": "Probability Distributions To Be Aware Of For Data Science (With Code) | Towards Data Science", "body_markdown": "![Revisit consent button](https://cdn-cookieyes.com/assets/images/revisit.svg)\nWe value your privacy\nWe use cookies to enhance your browsing experience, serve personalised ads or content, and analyse our traffic. 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This means it's essential for you to have a good grasp of some fundamental terminologies, what they mean, and how to identify them.\nOne such term you'll hear thrown around a lot is '_distribution_.' All this is in reference to is the properties of the data. There's several instances of phenomena in the real world that are considered to be statistical in nature (i.e. weather data, sales data, financial data, etc). This means there are several instances in which we've been able to develop methodologies that help us model nature through mathematical functions that can describe the characteristics of the data.\n> \"[...] a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.\"\n> * [Wikipedia](https://en.wikipedia.org/wiki/Probability_distribution)\n> \n\nKnowing the distribution of data helps us better model the world around us. It helps us to determine the likeliness of various outcomes, or make an estimate of the variability of an occurrence. All of this makes knowing different probability distributions extremely valuable in data science & machine learning.\nLearn this step by step with the interactive [AI Engineer](https://roadmap.sh/ai-engineer), [AI and Data Scientist](https://roadmap.sh/ai-data-scientist) and [Computer Science](https://roadmap.sh/computer-science) roadmaps.\nIn this article, we are going to cover a few distributions and share some Python code to display them visually.\n## Uniform Distribution\nThe most straightforward distribution you're likely to come across is the _uniform distribution;_ The uniform distribution is a type of probability distribution in which all outcomes are equally likely. For example, the probability of landing on any number if we roll a fair dice is 1/6. This is a discrete uniform distribution.\n![Probability Mass Function \\(PMF\\) of a discrete uniform distribution; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1sq7bCI7uR5dVEQhWsgStWg.png)\n**Probability Mass Function (PMF) of a discrete uniform distribution; Image by Author.**\nHowever, not all uniform distributions are discrete - they can also be continuous. This means there's an infinite number of equally likely measurable values which can take any real value within a specified range. The probability density function (PDF) of a continuous uniform distribution between a and b is as follows:\n![Probability Density Function \\(PDF\\) of a continuous uniform distribution; Image by Author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1eHmBJ3D55yywxVzVnnsy3g.png)\n**Probability Density Function (PDF) of a continuous uniform distribution; Image by Author**\nLet's see how to code them both in Python:\n![The output of the code above: Discrete and Continuous uniform distributions in Python; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1bpgW9l1GZ4T0Yjy09K4-vQ.png)\n**The output of the code above: Discrete and Continuous uniform distributions in Python; Image by Author.**\n## Gaussian distribution\nThe Gaussian distribution is likely to be the distribution you'll be most familiar with. It has several names: some call it the **bell curve** since the graph of its probability looks like a bell, some call it **Gaussian distribution** after Carl Gauss who was the German mathematician that first described it, and others call it the **normal distribution** because early statisticians noticed its reoccurrence over and over again.\nThe probability density function of a normal distribution is as follows:\n![Probability Density Function \\(PDF\\) for normal distribution; Image by Author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1DRAnup2MtniEKRSsg53ClA.png)\n**Probability Density Function (PDF) for normal distribution; Image by Author**\nσ is the standard deviation and μ is the mean of the distribution. But it's important to note that in a normal distribution, the mean, mode, and median are all equal.\nWhen we plot normally distributed random variables, the curve is symmetric around the mean - half of the values are to the left and half to the right of the center. And, the total area under the curve is 1.\n![The output of the code above: Gaussian distribution in Python; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1E_GQwOZOCfg_SQDLepjKUw.png)\n**The output of the code above: Gaussian distribution in Python; Image by Author.**\nSomething known about the normal distribution is the empirical rule. The empirical rule informs us of the percentage of data that falls within a certain number of standard deviations from the mean. These percentages are:\n * 68% of the data falls within one standard deviation of the mean.\n * 95% of the data falls within two standard deviations of the mean.\n * 99.7% of the data falls within three standard deviations of the mean.\n\n\n## Lognormal distribution\n> A lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable _X_ is log-normally distributed, then _Y_ = ln(_X_) has a normal distribution . [**Source** : [Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution)]\nHere's the PDF of a lognormal distribution:\n![Probability Density Function \\(PDF\\) for lognormal distribution; Image by Author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/11MeCtwzm7BsHsp8HBcamRQ.png)\n**Probability Density Function (PDF) for lognormal distribution; Image by Author**\nA random variable that is lognormally distributed takes only positive real values. Consequently, lognormal distributions create curves that are right-skewed.\nLet's plot this in Python:\n![The output of the code above: Lognormal distribution in Python; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1lsXEjz3g3w53jqetIHOJgg.png)\n**The output of the code above: Lognormal distribution in Python; Image by Author.**\n## Poisson distribution\nThe Poisson distribution is named after a French mathematician called [Siméon Denis Poisson](https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson). It's a discrete probability distribution which means it counts occurrences that have finite outcomes - in other words, it's a count distribution. Thus, the Poisson distribution is used to show the number of times an event is likely to occur over a specified period.\nIf an event occurs with a fixed rate in time, then the probability of observing the number (n) of events in time may be described with the Poisson distribution. For example, a customer may arrive at a cafe at an average rate of 3 per minute. We can use the Poisson distribution to calculate the probability of arrival for 9 customers in 2 minutes.\nHere's the probability mass function formula to help:\n![Probability Mass Function \\(PMF\\) for Poisson distribution; Image by Author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1irTi0K3TCzjRzBVBvj9nCg.png)\n**Probability Mass Function (PMF) for Poisson distribution; Image by Author**\n**λ** is the event rate in a time unit - in our example, it's 3. _**k**_ is the number of occurrences - in our example, it's 9. We can use `Scipy` to solve our example problem.\nThe curve of a Poisson distribution is similar to a normal distribution and λ marks the peak. Let's plot one in Python to see how this looks visually.\n![The output of the code above: Poisson distribution in Python; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1s28e3ase8RNeH36VfvpBcw.png)\n**The output of the code above: Poisson distribution in Python; Image by Author.**\n## Exponential distribution\n> The exponential distribution is the probability distribution of the time between events in a Poisson point process. [**Source** : [Wikipedia](https://en.wikipedia.org/wiki/Exponential_distribution)]\nThe probability density function of an exponential distribution is as follows:\n![Probability Density Function \\(PDF\\) for Exponential distribution; Image by Author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/19hpkWdsrXX44ekjYqBvtkg.png)\n**Probability Density Function (PDF) for Exponential distribution; Image by Author**\n**λ** is the rate parameter and **x** is the random variable.\nLet's see how we can plot an exponential distribution in Python:\n![The output of the code above: Exponential distribution in Python; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1rr8etDDzu4YnvRvRDpXNXg.png)\n**The output of the code above: Exponential distribution in Python; Image by Author.**\n## Binomial distribution\nYou can think of the binomial distribution as the probability of success or failure in an experiment. Some may also describe it as the coin toss probability.\nHere's Wikipedia's definition:\n> the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success or failure. [**Source** : [Wikipedia](https://en.wikipedia.org/wiki/Binomial_distribution)]\nEssentially, the binomial distribution measures the probability of two events. The probability of one event occurs with the probability p and the other event occurs with the probability 1-p.\nThis is the formula for a binomial distribution:\n![The formula for Binomial distribution; Source: Google Page](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1lyrX3wboveBzMXqBpQBL-Q.png)\n**The formula for Binomial distribution; Source:**[**Google Page**](https://www.google.com/search?q=what+is+binomial+distribution&rlz=1C1CHBF_en-GBGB921GB921&oq=hat+is+binomia+&aqs=chrome.1.69i57j0i13l9.7048j0j4&sourceid=chrome&ie=UTF-8)\nLet's see it in Python:\n![The output of the code above: Binomial distribution in Python; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1huPbs1aAuVfAdM10bVYnHg.png)\n**The output of the code above: Binomial distribution in Python; Image by Author.**\n## Student's t distribution\n> Student's _t_ -distribution (or simply the _t_ -distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym \"Student.\" [**Source** : [Wikipedia](https://en.wikipedia.org/wiki/Student%27s_t-distribution)]\nThe PDF is as follows:\n![Probability Density Function \\(PDF\\) for Student's t distribution; Image by Author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1kuoXOji0UW0pBCo0HL9Bng.png)\n**Probability Density Function (PDF) for Student's t distribution; Image by Author**\n**n** is the free parameter called \"_degrees of freedom_ \" but you may also see it referred to as \"_d.o.f._ \" The t distribution gets closer to a normal distribution for high values of n.\n![The output of the code above: Student's t distribution in Python; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1MWIeiagZLoTQxShFrPhikA.png)\n**The output of the code above: Student's t distribution in Python; Image by Author.**\n## Chi-squared distribution\nThe chi-squared distribution is a special case of the gamma distribution; With _**k**_ degrees of freedom, the chi-squared distribution is the sum of the squares of _**k**_ for some independent standard normal random variables.\nHere's the PDF:\n![Probability Density Function \\(PDF\\) for Chi-squared distribution; Image by Author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/15zCztpD1wQgLwbBS5qHXfQ.png)\n**Probability Density Function (PDF) for Chi-squared distribution; Image by Author**\nIt's a popular probability distribution, commonly used in hypothesis testing and in the construction of confidence intervals.\nLet's draw some example plots in Python:\n![The output of the code above: Chi-squared distribution in Python; Image by Author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1bzxn1oCnOOnEZdznDqaeVw.png)\n**The output of the code above: Chi-squared distribution in Python; Image by Author.**\n* * *\nHaving a good command of statistics and probability is essential for data science. In this article, I've shown you a few distributions you may want to remember as it's likely you'll come across them at some point in your journey. Feel free to save and reference this article for future records. Feedback would be extremely helpful for me also.\n_Thanks for reading._\n**Connect with me:** [LinkedIn](https://www.linkedin.com/in/kurtispykes/) [Twitter](https://twitter.com/KurtisPykes)\nIf you enjoy reading stories like this one and wish to support my writing, consider [becoming a Medium member](https://kurtispykes.medium.com/membership). With a $5 a month commitment, you unlock unlimited access to stories on Medium. If you use [my sign-up link](https://kurtispykes.medium.com/membership), I'll receive a small commission.\nAlready a member? 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"attachments/2c27fdbe4973/images/000.jpg", "bytes": 68496, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1sq7bCI7uR5dVEQhWsgStWg.png", "alt": "Probability Mass Function (PMF) of a discrete uniform distribution; Image by Author.", "local_path": "attachments/2c27fdbe4973/images/001.png", "bytes": 1901, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1eHmBJ3D55yywxVzVnnsy3g.png", "alt": "Probability Density Function (PDF) of a continuous uniform distribution; Image by Author", "local_path": "attachments/2c27fdbe4973/images/002.png", "bytes": 2174, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1bpgW9l1GZ4T0Yjy09K4-vQ.png", "alt": "The output of the code above: Discrete and Continuous uniform distributions in Python; Image by Author.", "local_path": "attachments/2c27fdbe4973/images/003.png", "bytes": 21058, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1DRAnup2MtniEKRSsg53ClA.png", "alt": "Probability Density Function (PDF) for normal distribution; Image by Author", "local_path": "attachments/2c27fdbe4973/images/004.png", "bytes": 3749, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1E_GQwOZOCfg_SQDLepjKUw.png", "alt": "The output of the code above: Gaussian distribution in Python; Image by Author.", "local_path": "attachments/2c27fdbe4973/images/005.png", "bytes": 14823, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/11MeCtwzm7BsHsp8HBcamRQ.png", "alt": "Probability Density Function (PDF) for lognormal distribution; Image by Author", "local_path": "attachments/2c27fdbe4973/images/006.png", "bytes": 6497, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1lsXEjz3g3w53jqetIHOJgg.png", "alt": "The output of the code above: Lognormal distribution in Python; Image by Author.", "local_path": "attachments/2c27fdbe4973/images/007.png", "bytes": 23985, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1irTi0K3TCzjRzBVBvj9nCg.png", "alt": "Probability Mass Function (PMF) for Poisson distribution; Image by Author", "local_path": "attachments/2c27fdbe4973/images/008.png", "bytes": 2548, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1s28e3ase8RNeH36VfvpBcw.png", "alt": "The output of the code above: Poisson distribution in Python; Image by Author.", "local_path": "attachments/2c27fdbe4973/images/009.png", "bytes": 6539, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/19hpkWdsrXX44ekjYqBvtkg.png", "alt": "Probability Density Function (PDF) for Exponential distribution; Image by Author", "local_path": "attachments/2c27fdbe4973/images/010.png", "bytes": 5498, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1rr8etDDzu4YnvRvRDpXNXg.png", "alt": "The output of the code above: Exponential distribution in Python; Image by Author.", "local_path": "attachments/2c27fdbe4973/images/011.png", "bytes": 10550, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1lyrX3wboveBzMXqBpQBL-Q.png", "alt": "The formula for Binomial distribution; Source: Google Page", "local_path": "attachments/2c27fdbe4973/images/012.png", "bytes": 22199, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1huPbs1aAuVfAdM10bVYnHg.png", "alt": "The output of the code above: Binomial distribution in Python; Image by Author.", "local_path": "attachments/2c27fdbe4973/images/013.png", "bytes": 3909, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1kuoXOji0UW0pBCo0HL9Bng.png", "alt": "Probability Density Function (PDF) for Student's t distribution; Image by Author", "local_path": "attachments/2c27fdbe4973/images/014.png", "bytes": 7963, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2022/01/1Wm1ZtnqcuLhOAk8gQ_V6Cw.png", "alt": "", "local_path": "attachments/2c27fdbe4973/images/015.png", "bytes": 82572, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2020/10/1WJ8sFo38MmouRX6Jfj3UNA-scaled.jpeg", "alt": "", "local_path": "attachments/2c27fdbe4973/images/016.jpeg", "bytes": 758500, "downloaded": true }, { "src": "https://assets.insightmediagroup.io/media/wp-content/uploads/2021/08/1ye313kXPyLee5hvxf8JQKA-scaled.jpeg", "alt": "", "local_path": "attachments/2c27fdbe4973/images/017.jpeg", "bytes": 451341, "downloaded": true }, { "src": 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