Probability Distribution | Formula, Types, & Examples
A probability distribution is a mathematical function that describes the probability of different possible values of a variable. Probability distributions are often depicted using graphs or probability tables.
Common probability distributions include the binomial distribution, Poisson distribution, and uniform distribution. Certain types of probability distributions are used in hypothesis testing, including the standard normal distribution, Student’s t distribution, and the F distribution.
Table of contents
What is a probability distribution?
A probability distribution is an idealized frequency distribution.
A frequency distribution describes a specific sample or dataset. It’s the number of times each possible value of a variable occurs in the dataset.
The number of times a value occurs in a sample is determined by its probability of occurrence. Probability is a number between 0 and 1 that says how likely something is to occur:
- 0 means it’s impossible.
- 1 means it’s certain.
The higher the probability of a value, the higher its frequency in a sample.
More specifically, the probability of a value is its relative frequency in an infinitely large sample.
Infinitely large samples are impossible in real life, so probability distributions are theoretical. They’re idealized versions of frequency distributions that aim to describe the population the sample was drawn from.
Variables that follow a probability distribution are called random variables. There’s special notation you can use to say that a random variable follows a specific distribution:
- Random variables are usually denoted by X.
- The ~ (tilde) symbol means “follows the distribution.”
- The distribution is denoted by a capital letter (usually the first letter of the distribution’s name), followed by brackets that contain the distribution’s parameters.
For example, the following notation means “the random variable X follows a normal distribution with a mean of µ and a variance of σ2.”
There are two types of probability distributions:
Discrete probability distributions
A discrete probability distribution is a probability distribution of a categorical or discrete variable.
Discrete probability distributions only include the probabilities of values that are possible. In other words, a discrete probability distribution doesn’t include any values with a probability of zero. For example, a probability distribution of dice rolls doesn’t include 2.5 since it’s not a possible outcome of dice rolls.
The probability of all possible values in a discrete probability distribution add up to one. It’s certain (i.e., a probability of one) that an observation will have one of the possible values.
A probability table represents the discrete probability distribution of a categorical variable. Probability tables can also represent a discrete variable with only a few possible values or a continuous variable that’s been grouped into class intervals.
A probability table is composed of two columns:
- The values or class intervals
- Their probabilities
|“Salutations, organic life-form.”||.2|
Notice that all the probabilities are greater than zero and that they sum to one.
Probability mass functions
A probability mass function (PMF) is a mathematical function that describes a discrete probability distribution. It gives the probability of every possible value of a variable.
A probability mass function can be represented as an equation or as a graph.
Common discrete probability distributions
|Binomial||Describes variables with two possible outcomes. It’s the probability distribution of the number of successes in n trials with p probability of success.||The number of times a coin lands on heads when you toss it five times|
|Discrete uniform||Describes events that have equal probabilities.||The suit of a randomly drawn playing card|
|Poisson||Describes count data. It gives the probability of an event happening k number of times within a given interval of time or space.||The number of text messages received per day|
Continuous probability distributions
A continuous probability distribution is the probability distribution of a continuous variable.
A continuous variable can have any value between its lowest and highest values. Therefore, continuous probability distributions include every number in the variable’s range.
The probability that a continuous variable will have any specific value is so infinitesimally small that it’s considered to have a probability of zero. However, the probability that a value will fall within a certain interval of values within its range is greater than zero.
Probability density functions
A probability density function (PDF) is a mathematical function that describes a continuous probability distribution. It provides the probability density of each value of a variable, which can be greater than one.
A probability density function can be represented as an equation or as a graph.
In graph form, a probability density function is a curve. You can determine the probability that a value will fall within a certain interval by calculating the area under the curve within that interval. You can use reference tables or software to calculate the area.
The area under the whole curve is always exactly one because it’s certain (i.e., a probability of one) that an observation will fall somewhere in the variable’s range.
A cumulative distribution function is another type of function that describes a continuous probability distribution.
Common continuous probability distributions
|Normal||Describes data with values that become less probable the farther they are from the mean, with a bell-shaped probability density function.||SAT scores|
|Continuous uniform||Describes data for which equal-sized intervals have equal probability.||The amount of time cars wait at a red light|
|Log-normal||Describes right-skewed data. It’s the probability distribution of a random variable whose logarithm is normally distributed.||The average body weight of different mammal species|
|Exponential||Describes data that has higher probabilities for small values than large values. It’s the probability distribution of time between independent events.||Time between earthquakes|
How to find the expected value and standard deviation
You can find the expected value and standard deviation of a probability distribution if you have a formula, sample, or probability table of the distribution.
The expected value is another name for the mean of a distribution. It’s often written as E(x) or µ. If you take a random sample of the distribution, you should expect the mean of the sample to be approximately equal to the expected value.
- If you have a formula describing the distribution, such as a probability density function, the expected value is usually given by the µ parameter. If there’s no µ parameter, the expected value can be calculated from the other parameters using equations that are specific to each distribution.
- If you have a sample, then the mean of the sample is an estimate of the expected value of the population’s probability distribution. The larger the sample size, the better the estimate will be.
- If you have a probability table, you can calculate the expected value by multiplying each possible outcome by its probability, and then summing these values.
What is the expected value of robin eggs per nest?
Multiply each possible outcome by its probability:
|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|
Sum the values:
E(x) = 0.4 + 1.5 + 1.2
E(x) = 3.1 eggs
The standard deviation of a distribution is a measure of its variability. It’s often written as σ.
- If you have a formula describing the distribution, such as a probability density function, the standard deviation is sometimes given by the σ parameter. If there’s no σ parameter, the standard deviation can often be calculated from other parameters using formulas that are specific to each distribution.
- If you have a sample, the standard deviation of the sample is an estimate of the standard deviation of the population’s probability distribution. The larger the sample size, the better the estimate will be.
- If you have a probability table, you can calculate the standard deviation by calculating the deviation between each value and the expected value, squaring it, multiplying it by its probability, and then summing the values and taking the square root.
|Eggs (x)||Probability (P(x))||x – E(x)|
|2||.2||2 − 3.1 = −1.1|
|3||.5||3 − 3.1 = −0.1|
|4||.3||4 − 3.1 = 0.9|
Square the values and multiply them 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|
Sum the values and take the square root:
σ = √(0.242 + 0.005 + 0.243)
σ = √(0.49)
σ = 0.7 eggs
How to test hypotheses using null distributions
All hypothesis tests involve a test statistic. Some common examples are z, t, F, and chi-square. A test statistic summarizes the sample in a single number, which you then compare to the null distribution to calculate a p value.
The p value is the probability of obtaining a value equal to or more extreme than the sample’s test statistic, assuming that the null hypothesis is true. In practical terms, it’s the area under the null distribution’s probability density function curve that’s equal to or more extreme than the sample’s test statistic.
|Standard normal (z distribution)||One-sample location test|
|Student’s t distribution||One-sample t test
Two-sample t test
Paired t test
Comparison of nested linear models
Equality of two variances
|Chi-square||Chi-square goodness of fit test
Test of a single variance
Probability distribution formulas
|Distribution||Formula||Type of formula|
|Binomial||Probability mass function|
|Discrete uniform||Probability mass function|
|Poisson||Probability mass function|
|Normal||Probability density function|
|Continuous uniform||Probability density function|
|Exponential||Probability density function|
Frequently asked questions about probability distributions
- What’s the difference between relative frequency and probability?
Probability is the relative frequency over an infinite number of trials.
For example, the probability of a coin landing on heads is .5, meaning that if you flip the coin an infinite number of times, it will land on heads half the time.
Since doing something an infinite number of times is impossible, relative frequency is often used as an estimate of probability. If you flip a coin 1000 times and get 507 heads, the relative frequency, .507, is a good estimate of the probability.
- What is a normal distribution?
- What are the two types of probability distributions?