Probability Distributions: Used in Data Science
If you’ve ever asked, “What’s the chance that this happens?”, you’ve touched on the essence of probability distributions. In data science, distributions are maps that show you where the outcomes are likely to occur. They’re used in A/B testing, fraud detection, user churn prediction, tuning machine-learning models… You name it, distributions are most likely a part of it. The hard part is that distributions can seem abstract until you see one in action.
This guide, made simple, turns that abstract mathematical content into tangible intuition. We’ll discuss what probability distributions are, the types of probability distributions that you’re likely to encounter, what a probability distribution function (PDF/PMF) is and does, and the applied, real-world uses of probability distributions in data science and computer science. You’ll get nice formulas, explanations in plain language, and short examples you can pull for your own work. Stay until the end for two fully solved questions that demonstrate how the math works out, step-by-step.
Table Of Content
What is a Probability Distribution?
Core Tools in Probability Distributions
The Types of Probability Distribution
Uses of Probability Distributions in Data Science
Conclusion
Frequently Asked Questions
What is a Probability Distribution?
A probability distribution is like a map that lays out how likely each possible outcome of some random event is. It’s a way of assigning probabilities to all the possible values a random variable can take.
You can think of a random variable as taking on different values at random, like rolling a dice, recording someone’s height, or counting how many emails you receive in one day.
Core Tools in Probability Distributions
PMF (Probability Mass Function) – Used for discrete variables.
- It tells you the probability of getting an exact value.
- Example: If XXX is the number of heads in 3 coin tosses, PMF gives P(X=2)P(X = 2)P(X=2).
PDF (Probability Density Function) – Used for continuous variables.
- It gives a “density,” not a direct probability.
- You find probability by calculating the area under the curve between two values.
- Formula:
P(a≤X≤b)=∫abf(x) dxP(a \leq X \leq b) = \int_a^b f(x) \, dxP(a≤X≤b)=∫abf(x)dx
Example: Probability that a person’s height is between 160 cm and 170 cm.
CDF (Cumulative Distribution Function) – Works for both discrete and continuous variables.
- It tells you the probability that XXX is less than or equal to a certain value.
- Formula:
F(x)=P(X≤x)F(x) = P(X \leq x)F(x)=P(X≤x) - Example: Probability that a student scores 70 or less on a test.
The Types of Probability Distribution
Given below are 9 major types of probability distributions with formulas:
1. Normal Distribution
This type of probability distribution, depicted using a symmetrical bell-shaped curve, the normal distribution, is frequently observed in nature and society. For instance, IQ scores in a population typically follow the normal distribution, which can help educators and psychologists understand the direction of overall intelligence. And subsequently, help tailor and design continued learning.
Data-science use: Z-scores, confidence intervals, and error terms in regression (assumptions).
The formula for the probability density function (PDF) of a normal distribution is:
f(x) = (1 / (σ * √(2π))) * e^(-(x – μ)² / (2σ²))
Where:
- f(x) is the probability density at a given value x.
- μ (mu) is the mean (average) of the distribution.
- σ (sigma) is the standard deviation, a measure of the spread of the distribution.
- π (pi) is approximately 3.14159.
- e is Euler’s number, approximately 2.71828.
2. Binomial Distribution
The binomial distribution describes the probability of observing a fixed number of successes in a fixed number of independent trials with a fixed success probability. For example, you might flip a coin 10 times and want to know how many times you got heads.
For example, if a basketball player makes free throws (with a 70% success rate), we can use a binomial distribution to find the probability of actually making exactly 7 out of 10 shots.
The binomial distribution formula is for any random variable X, given by;
P(x : n, p) = (n choose x) × (p^x) × (1 – p)^(n – x)
or
P(x : n, p) = (n choose x) × (p^x) × (q)^(n – x), where q = 1 – p
Where,
n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of Success in a single experiment
q = Probability of Failure in a single experiment = 1 – p
3. Bernoulli Distribution
The simplest of them all, the Bernoulli distribution models binary outcomes: success or failure, yes or no, win or lose. Tossing a coin, where the result is either heads (success) or tails (failure), is a classic example.
This distribution plays a role in quality control, where each product is judged as either meeting the standard or not. With only two outcomes, it’s an essential model for binary decision-making processes.
For the Bernoulli Distribution, it helps in finding the probability of getting a success (1) or a failure (0).
P(X=x) = px(1-p)1-x, x = 0, 1; 0 < p < 1
Here,
- x can only be 0 or 1.
- The PDF, which is denoted as P(X = x), calculates the probability that the random variable X equals a specific value x.
- If x=1 (success), then the probability is p.
- If x=0 (failure), then the probability is q, which is the complement of p and can also be written as 1-p.
4. Poisson Distribution
This distribution expresses the likelihood of a certain number of events happening in a fixed time or space period, assuming they occur at a fixed rate. An example of this is a call center that may want to evaluate the number of customer calls per hour and model it with a Poisson distribution.
This situation is perfect for modeling something that happens randomly but maintains an average with an underlying constant average, for example, requests for customer service, traffic accidents at a given intersection a day, or emails received a day.
Poisson distribution formula
The probability mass function of the Poisson distribution is:
Where:
- is a random variable following a Poisson distribution
- is the number of times an event occurs
- ) is the probability that an event will occur k times
- is Euler’s constant (approximately 2.718)
- is the average number of times an event occurs
- ! is the factorial function
5. Exponential Distribution
Often used to model the time until an event occurs, the exponential distribution is common in reliability engineering. It can represent how long a machine operates before breaking down, helping schedule maintenance before failures happen.
It also models natural phenomena such as the time intervals between earthquakes in a given region, assuming a consistent average occurrence rate.
Exponential Distribution Formula
The continuous random variable, say X, is said to have an exponential distribution if it has the following probability density function:
Where
λ is called the distribution rate.
6. Gamma Distribution
The gamma distribution is an extension of the exponential distribution and describes the sum of several exponential variables. It is widely used in queuing theory (for example, estimating the wait time before being served using several customers in line).
For example, if customers arrive in a queue using the Poisson process, and their service times are distributed exponentially, we can use the gamma distribution to estimate the total time needed to serve a certain number of customers.
Gamma Distribution Formula
Where p and x are continuous random variables.
7. Beta Distribution
The beta distribution is ideal to model probabilities and proportions, since it always occurs between 0 and 1. For example, in marketing, analysts use beta distribution to measure the conversion rate of a website and ultimately improve user-engagement strategies.
The beta distribution can also help in A/B testing by measuring uncertainty around conversion types between two versions of a webpage. This allows businesses to look at the different design decisions through a data-driven lens.
In data science, the beta distribution is defined by the following probability density function (PDF): f(x; α, β) = (x^(α-1) * (1-x)^(β-1)) / B(α, β) for 0 ≤ x ≤ 1, where α and β are shape parameters, and B(α, β) is the beta function
8. Uniform Distribution
When outcomes have an equal probability of occurring, we have a uniform distribution. Rolling a fair six-sided die, where each face represents a 1/6 chance of appearing, is the classic example.
This idea is used in simulations, games, and randomized experiments where aspects of fairness and equal chance are important.
9. Log-Normal Distribution
A log-normal distribution is related to variables when the logarithms of the variables are normally distributed. In finance, the prices of stocks can be well described by the log-normal distribution, which allows for the estimation of price movement by the investor.
Economics begins to warrant consideration here as well; the distribution of wealth may show log-normal characteristics, with everyone being at a middle income and fewer people at the far ends of wealth extremes.
The lognormal distribution is a two-parameter distribution with parameters μ and σ. The probability density function can be defined as:
Here, t values are the time-to-failure.
Mean of the natural logarithms of the time-to-failure.
Standard deviation (SD) of the natural logarithms of the time-to-failure.
Uses of Probability Distributions in Data Science
Probability distributions are fundamental in data science since they enable a model of uncertainty, quantify risk, and support decision-making. Probability distributions allow for the development of realistic models that incorporate the randomness seen in observations from real-world data. Given below are the common uses of probability distribution:
A/B Testing
The Beta-Binomial and other Bayesian models are excellent at measuring conversion rates and so that we know the uncertainty. A standard z-test, whose five assumptions of normal distribution are also necessary to get estimates, if the performance differences are real.
Forecasting and Anomaly Detection
Count data can be modeled with the Poisson distribution (e.g., logins to a website per hour), and normal distributions can represent the residuals of the time series forecast; this helps me indicate that a website’s visits are not as expected and whether it is an anomaly or just part of the data’s randomness.
Reliability and Survival Analysis
Hazard models are based on exponential, gamma, or Weibull distributions, and quantify time-to-failure data that helps predict the age at which machines, systems, or products are likely to break down.
Natural Language Processing (NLP)
The multinomial distribution encodes word counts and frequencies for applications like text classification and topic modeling, and also contributes to probabilistic language models.
Computer Vision
Gaussian distributions model noise from sensors/environments, while a log-normal distribution can be used to model illumination variability across images.
Risk modeling
Log-normal distributions can incorporate the skewness typical in modeling asset prices, and the Poisson distribution can be used to model claim frequencies in insurance modelling.
Conclusion
Probability Distributions are common in many fields, including insurance, physics, engineering, computer science, and some social sciences, where psychology and medical students are using probability distributions widely. It has a simple use and it is used in many ways. This article has described and analyzed six important distributions identified in everyday life. You should now be able to identify, relate, and distinguish between these distributions.
Frequently Asked Questions
PMF applies to discrete data, giving the exact probability of each outcome. PDF applies to continuous data, showing probability density—actual probabilities come from the area under the curve. CDF works for both, giving the probability that a value is less than or equal to a given point.
Identify if data is discrete or continuous, compare patterns with candidate distributions, and validate using plots or statistical tests.
They power A/B testing, forecasting, anomaly detection, and more.
Use methods like moments, maximum likelihood estimation, or Bayesian inference, then validate the fit.
