Probability Distribution – Formula, Calculations & Examples

25.04.23 Statistics Time to read: 5min

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If you have several potential results from an experiment, you can use a probability distribution function to estimate each possibility. When describing it, the probability density function and the cumulative distribution function stand out as particularly useful.1

In this article, we will explore probability distribution and all its connected features.

Probability Distribution – In a Nutshell

  • Two forms of data may be used in statistical analysis: discrete and continuous.
  • One way to measure the possibility of something happening is to look at its probability distribution function.
  • The parameters of a probability distribution specify its form.

Definition: Probability distribution

Probability distributions are mathematical functions characterizing the likelihood that certain variable values will be attained. Graphs and probability tables are common ways to visualize probability distributions.

Example of probability distribution:

A probability table may be used to characterize the range of outcomes from a single coin toss:

Outcome Probability
Heads Tails
.5 .5

Binomial, Poisson, and uniform distributions are all examples of popular types of probability distributions.

The ordinary, normal distribution, the F distribution, and the student’s t distribution are all employed in hypothesis testing.

Probabilities of distributions are used to characterize the populations of real-world variables, such as the outcomes of coin flips and the masses of chicken eggs.2 P values are also calculated from them when conducting hypothesis tests.

Example of a probability distribution idealized as a frequency distribution:

Consider a farmer interested in the odds of a specific size egg being produced by her farm. The farmer takes 100 eggs at random and uses a histogram to explain the distribution of the eggs’ weights.

The most reasonable approach is acknowledging that the size of eggs seems to follow a normal, standard distribution.

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Discrete probability distribution

Discrete likelihood distributions are variables that can only take on one of many possible values. Distributions of discrete probabilities only contain those values for which a possibility can be calculated.

Probability tables

Probability tables are used to graphically depict the discrete distribution of categorical data. Two columns (class limits) and (their probabilities) make up the table.


A robotic voice offers an unexpected welcome to humans. Below is a probability table that describes the distribution of greetings given certain conditions:

Greetings Probability
Hi! .2
Howdy! .1
Greetings, Human! .7

Probability mass function (PMF)

Discrete probability distributions may be modeled by means of a mathematical function known as a probability mass function (PMF). Essentially, it calculates the likelihood of each potential value for a variable.


Consider the possibility that the average UK shirt collection follows a Poisson distribution. The formula gives the PMF:

P(X=k) is the likelihood that one has exactly k shirts.
Λ is the average amount of sweaters owned by a population.
E is Euler’s constant

Types of discreet probability distributions

Distribution Description Example
Binominal Identifies factors with two distinct results. The probability that, after five flips, a coin will come up heads every time.
Discrete uniform Defines situations when the odds of happening are the same for everyone. The typical distribution of playing card suits.
Poisson It calculates the likelihood of k events occurring in a given period or place. The daily number of texts.

Continuous probability distributions

A continuous variable admits an endless number of possible values. When a variable is considered continuous across a non-empty subset of the real numbers, it may take on any value inside that subset.

Probability density functions (PDF)

PDFs are mathematical functions used to characterize continuous probability distributions. As such, it gives us the probability density for all possible values of a variable, even those that exceed one.

It is possible to depict a probability density function as an equation or a graph.

A curve represents the probability density function when plotted on a graph. Determining the area under the curve within a certain range might help you estimate the likelihood that a given value will lie within that gap.3 Another sort of function used to characterize continuous probability distributions is the cumulative distribution function.

Example of probability density function:

A regular distribution of egg weight may be described using the formula:

f(x) is the distribution of egg weight probabilities.
μ is the average weight of a group's eggs (1.85 oz).
σ defines the population's variation in egg weight (0.13 oz).

It is statistically impossible for an egg to weigh precisely 2 ounces. A typical scale would say an egg weighs 2 ounces, but an infinitely exact scale might reveal minimal discrepancy.

Types of continuous probability distributions

With continuous data, many alternative distribution functions are possible. This set of handouts features:

Distribution Explanation Example
Exponential Data is described in which a small value's likelihood is more significant than a great value. Time interval distribution is the likelihood of occurrence between unrelated events. Intervals between earthquakes
Continuous uniform Describes information when intervals of similar size have the same probability. Time spent at a green light by vehicles.
Normal distribution Provides a bell-shaped density function of the likelihood for data, where outlying numbers become less likely as the distance from the mean increases. Analyzing SAT results
Log-normal Simulates right-skewed patterns, often occurring when growth rates are assumed constant regardless of initial sample size. Percentage information about body fat

Probability distribution: Expected value & standard deviation

Obtaining an equation, data or probability chart for a distribution allows one to calculate its estimated return and standard deviation. A distribution’s mean is also known as its anticipated value. As an expression, it is often represented as E(x).

Expected value

Hyalines usually have a nest size of two to four eggs. Consider the following hypothetical probability distribution of Hyaline egg counts per nest.

Eggs (x) Probability (P(x)) x × P(x)
2 .2 2 × 0.2= 0.4
3 .5 3 × 0.5= 1.5
4 .3 4 × 0.3= 1.2

The expected value of eggs per net is the sum of all values after multiplying the number of eggs and probability.

E(x)= 0.4+1.5+1.2= 3.1 eggs

Standard deviation

To find the standard deviation, compute every value’s deviation from the expected value. Then multiply the squared values by their probability:

Eggs (x) Probability (P(x)) x - E(x) [x – E(x)] 2 × P(x)
2 .2 2-3.1= -1.1 (-1.1) 2 × 0.2 = 0.242
3 .5 3-3.1= -0.1 (-0.1) 2 × 0.5 = 0.005
4 .3 4-3.1= 0.9 (0.9) 2 × 0.3 = 0.243

Obtain the square root after adding the values.

√0.242 + 0.005 + 0.243= 0.7

Probability distribution: Testing hypotheses with null distributions

Evaluating hypotheses often makes use of null distributions. When the test’s null hypothesis is accepted as true, a statistic’s likelihood falls inside a range called its null distribution.4 The sample is summed up in a single number called a test statistic, which is then compared to the null distribution to get a probability value.

If the null hypothesis is true, the p-value represents the likelihood of receiving a number equal to or more drastic than the sample’s statistical test.

Testing hypotheses using null distributions

One-sample t-tests are hypothesis tests that use the usage of the test statistic known as Student’s t. In the null distribution of Student’s t, the p-value (for a one-sided test) is the shading region to the right of t = 1.7.

The area is equal to.06 and may be determined using a calculator, statistical software, or reference tables. As this is a rather small sample, the significance level is.06.

Common null distributions

Distribution Statistical test
F- distribution ANOVA
Equality of two variances
Chi-square Chi-square goodness of fit test
Test of a single variance
Standard normal One sample location test
Student's t distribution Linear regression
One sample t-test.
Two sample t-tests
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Probability distribution: Formulas

In the following, the formulas of the probability distribution will be listed:

Distribution Formula Type of formula
Normal Probability density function
Poisson Probability mass function
Discrete Uniform Probability mass function
Binominal Probability mass function


Probability distributions are mathematical functions that characterize the likelihood of a set of alternative values for a variable.

A toss of a coin is the most basic illustration. There are two potential results when flipping a coin:

  1. Heads
  2. Tails

The data in a normal distribution are spread out evenly throughout the board. There seems to be a core zone where most values are concentrated, with a gradual decrease in value as one moves outside that area.

There are two main types of probability distributions:

  1. Continuous
  2. Discrete


1 Shinde, Siddhesh. “What Is Probability Distribution? How Does One Use It in Real Life?.” Emeritus Online Courses. February 16, 2023.

2 University of North Florida. “Distributions.” Accessed April 20, 2023.

3 BYJUS. “Normal Distribution Formula in Probability and Statistics.” Accessed April 20, 2023.

4 Iowa State University. “Random variables and probability distributions.” Accessed April 20, 2023.