DISCRETE AND CONTINUOUS UNIFORM DISTRIBUTIONS

Let’s understand about the uniform distributions:

Uniform Distribution is divided into two types

1. Discrete Uniform Distribution: If a random variable is discrete and follows uniform distribution then it is a discrete uniform distribution.
2. Continuous Uniform Distribution: If a random variable is continuous and follows uniform distribution then it is continuous uniform distribution.

DISCRETE UNIFORM DISTRIBUTION

Probability Mass Function(PMF) for discrete random variables.

In Uniform distribution all values are equally probable.

Let’s Understand with the help of example:

Example:

Let us assume there is a fair dice 1,2,3,4,5,6 are all probable number with probability of 1/6 for each number.

a = 1, b = 6, n = 6

and probability of getting any number is 1/n = 1/6

U(a,b) → Uniform Distribution where b > a.

n = b-a+1

Fun part is that, if we know a and b we know everything about the distributions

Uniform distribution says that the probability of getting any value is uniform and is exactly the same.

In PMF, we can only represent by dots.

Probability of getting the value of 1 is same as probability of getting the value of any number 2,3,4,5,6.

There is no probability of getting any other value in between 2 and 3 and so on. so it is called discrete random variables.

Cummulative Distribution Function is a non-smooth function in this case

U(a,b)

n = b-a+1

PMF = 1/n

Mean = (a+b)/2

Median = (a+b)/2

Variance = (b-a+1)²-1/12

Skewness = 0

Credit- wikipedia

CONTINUOUS UNIFORM DISTRIBUTION

Probability Density Function(PDF) is for continuous random variable

U(a,b) is the uniform distribution where a < b

a is minimum value

b is maximum value

All the values in this interval

PDF of a uniform distribution of a continuous random variable is a continuous line.

The probability of any value is the same 1/(b-a)

The area under probability curve/PDF should be 1

PDF = 1/(b-a) &a < b

Mean = (a+b)/2

Median = (a+b)/2

Credit- wikipedia