Difference between Continuous and Discrete Uniform Distribution
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,…,xn | F(x)=Number of outcomes ≤ 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,…,xn | 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,
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].