Uniform Probability Distribution Learning

Introduction to uniform probability distribution learning:

In mathematics, uniform distribution is one interesting topics in probability theory and statistics. A continuous random variable X are said to be follow a uniform distribution, if it assumes only a values between a to b and its probability density function is represented by

f(x) =`1/(b-a)` , for a < x < b

Otherwise, f(x) = 0, then the values of x is not in the interval a and b.

In this article, we shall learning about the uniform probability distribution. The following are the example problems in uniform probability distribution learning.

Uniform Probability Distribution Learning - Formulas:

Some of the uniform probability distribution formulas for calculating mean, variance and standard deviation are

Mean (or) Expected value:

E(x) = `(b + a)/2`

Variance:

σ2 = `((b - a)^2)/12`

Standard deviation:

σ = `sqrt(((b-a)^2)/12)`

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Uniform Probability Distribution Learning - Example Problems:

Example 1:

If X is a continuous random variable for uniform probability distribution in the interval of 0 < x < 25. Estimate the expected value, variance and standard deviation.

Solution:

Given

0 < x < 25

f(x) = `1/(b -a)`

Here a = 0 and b = 25

f(x) = `1/(25 - 0)`

= `1/25`

f(x) = `1/25`

Expected value (or) Mean:

E(x) = `(b + a)/2`

= `(25 + 0)/2`

= `25/2`

= 12.5

E(x) = 12.5

Variance:

σ2 = `((b - a)^2)/12`

= `((25 - 0)^2)/12`

= `(25^2)/12`

= `625/12`

= 52.08

σ2 = 52.08

Standard deviation:

σ = `sqrt(((b -a)^2)/12)`

= `sqrt(((25 - 0)^2)/12)`

= `sqrt((25^2)/12)`

= `sqrt(625/12)`

= `sqrt(52.08)`

= 7.22

σ = 7.22

Answer:

f(x) = `1/25`

Mean (or) Expected value E(x) = 12.5

Variance σ2 = 52.08

Standard deviation σ = 7.22

Example 2:

If X is a continuous random variable for uniform probability distribution in the interval of 1 < x < 15. Estimate the expected value, variance and standard deviation.

Solution:

Given

1 < x < 15

f(x) = `1/(b -a)`

Here a = 1 and b = 15

f(x) = `1/(15 - 1)`

= `1/14`

f(x) = `1/14`

Expected value (or) Mean:

E(x) = `(b + a)/2`

= `(15 + 1)/2`

= `16/2`

= 8

E(x) = 8

Variance:

σ2 = `((b - a)^2)/12`

= `((15 - 1)^2)/12`

= `(14^2)/12`

= `196/12`

= 16.33

σ2 = 16.33

Standard deviation:

σ = `sqrt(((b -a)^2)/12)`

= `sqrt(((15 - 1)^2)/12)`

= `sqrt((14^2)/12)`

= `sqrt(196/12)`

= `sqrt(16.33)`

= 4.04

σ = 4.04

Answer:

f(x) = `1/14`

Mean (or) Expected value E(x) = 8

Variance σ2 = 16.33

Standard deviation σ = 4.04