Continuous Vs Discrete Uniform Distribution

Last Updated : 05 Jun, 2024

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Continuous and discrete uniform distributions are two types of probability distributions. A continuous uniform distribution has an interval of equally likely values. Instead, a discrete uniform distribution applies to a finite set of outcomes with equal probabilities.

In this article, we will discuss continuous and discrete uniform distribution along with a difference between them.

Table of Content

What is Continuous Distribution?
What is Discrete Uniform Distribution?
Formula of Continuous and Discrete Uniform Distribution
Difference between Continuous and Discrete Uniform Distribution

What is Continuous Distribution?

Continuous uniform distribution is a probability distribution in which all outcomes are equally likely within a specified interval [a,b]. In other words, the probability density function (PDF) is constant over this interval, and the distribution is defined by the two parameters a and b, which are the lower and upper bounds, respectively.

In a continuous distribution:

The total area under the probability density function (PDF) curve for a continuous distribution is equal to 1.
Examples of continuous distributions include normal, uniform, and exponential distributions.

Example of Continuous Distribution

Examples of continuous uniform distribution are:

Random Time of Day
Random Position on a Line Segment
Random Point in a Unit Square

Properties of Continuous Uniform Distribution

The properties of continuous uniform distribution are:

Symmetry: The distribution is perfectly symmetric about the mean.

Bell-shaped Curve: The shape of the normal distribution is often referred to as a “bell curve”.

Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.

What is Discrete Uniform Distribution?

Discrete uniform distribution is a kind of probability distribution in which every possible result has equal likelihood of occurrence. When there are limited possibilities and every one of them is equally likely, this distribution is applied.

Example of Discrete Uniform Distribution

Examples of discrete uniform distribution are:

Rolling a Fair Die
Drawing a Card from a Deck
Choosing a Random Day of the Week
Flipping a fair coin

Properties of Discrete Uniform Distribution

The properties of discrete uniform distribution are:

Equal Probability: Each outcome has an equal chance of occurring.

Finite Set of Outcomes: The possible outcomes are discrete and finite.

Uniformity: The probability mass function (PMF) is constant over the range of possible outcomes.

Formula of Continuous and Discrete Uniform Distribution

Below are formulas of continuous and discrete uniform distribution:

Distribution Type	Description	Probability Density Function/ Probability Mass Function	Cumulative Distribution Function
Discrete Uniform	Finite set of equally likely outcomes	P(X=x)= 1/n for x=x₁,x₂,…,x_n	F(x)=Numberofoutcomes ≤ x/n
Continuous Uniform	Continuous range of equally likely outcomes between a and b	f(x)= 1/(b-a) for a ≤ x ≤ b	F(x)= [Tex]\begin{cases} 0 & \text{if } x < a \\ \frac{x – a}{b – a} & \text{if } a \leq x \leq b \\ 1 & \text{if } x > b \end{cases}[/Tex]

Difference between Continuous and Discrete Uniform Distribution

The difference between continuous distribution and discrete uniform distribution can be understood from the table given below.

Basis	Discrete Uniform Distribution	Continuous Distribution
Nature of Outcomes	Finite and countable set of outcomes	Infinite and uncountable range of outcomes
Probability Function	Probability Mass Function (PMF): P(X=x)= 1/n	Probability Density Function (PDF): f(x) = 1/(b-a)
Range of Values	Specific discrete values x₁,x₂,…,x_n	Continuous range of values between a and b
Probability Calculation	Equal probability for each outcome: P(X=x)= 1/n	Equal density across the interval: f(x)= 1/(b-a)
Cumulative Distribution	CDF increases stepwise with each outcome and is defined by F(x) = P(X ≤ x).	CDF is a linear function within the interval defined by F(x) = (x – a) / (b – a) for a ≤ x ≤ b
Support	Specific values within a finite set	Continuous interval [a,b]
Real-World Application	Games of chance, like dice rolls or card draws	Random selection within a time interval, length measurement, etc.
Example	Rolling a fair six-sided die (outcomes: 1, 2, 3, 4, 5, 6)	Selecting a random point on a line segment from 1 to 10

Also Check,

Difference Between Variance and Standard Deviation
Correlation vs Causation
CDF vs. PDF

FAQs on Continous and Discrete Uniform Distribution

Is uniform distribution a continuous distribution?

A uniform distribution can be continuous if it describes a range of values with equal probability within a specified interval. It is called a continuous uniform distribution. There is also a discrete uniform distribution for finite sets of equally likely outcomes.

Is normal distribution discrete or continuous?

A normal distribution is continuous. It describes a continuous range of values where data is symmetrically distributed around the mean, forming a bell-shaped curve.

Is poisson distribution discrete or continuous?

A Poisson distribution is discrete. It describes the probability of a given number of events occurring in a fixed interval of time or space.

Is Bernoulli distribution discrete or continuous?

A Bernoulli distribution is discrete. It describes a random experiment with exactly two possible outcomes: success (1) or failure (0).

What is the uniform distribution of discrete variables?

The uniform distribution of discrete variables, also known as the discrete uniform distribution, is a probability distribution where each of a finite number of outcomes is equally likely.

What is the uniform distribution of continuous variables?

The uniform distribution of continuous variables, also known as the continuous uniform distribution, is a probability distribution in which all outcomes are equally likely within a specified interval [a,b].

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