Continuous Uniform Distribution

Uniform Distribution

The simplest continuous probability distribution is the uniform distribution. It is observed in many situations. In general, when the frequencies of various classes are observed to be more or less equal, the situation is the best for uniform distribution. It has prime importance in the field of simulation. In many places, uniform distribution is useful to develop further theory. The distribution function F(x) of a continuous random variable follows uniform distribution on the interval [0, 1], irrespective of the distribution of X. This property is useful in drawing model samples or simulation.

Uniform or Rectangular Distribution

In this distribution, the p.d.f. of the r.v. remains constant over the range space of the variable.

Definition

A continuous type r.v. X is said to follow uniform distribution over interval [a, b], if its p.d.f. is given by,

f(x) = 1 / (b - a), a ≤ x ≤ b

f(x) = 0, otherwise

Notation: X ~ U [a, b]

The distribution is also known as 'rectangular distribution', as the graph of p.d.f. describes a rectangle over the X-axis and between the ordinates at X = a and X = b.

Note that:

1. f(x) = 0 for all x ∉ R
2. ∫_a^b f(x) dx = 1

Properties

The p.d.f. of uniform distribution is constant over its support [a, b]. The p.d.f. is reciprocal of the length of interval in which it takes places.

Nature of p.d.f. curve: It is flat over its range.

If X ~ U [-a, a], then

f(x) = 1 / (2a), -a ≤ x ≤ a

f(x) = 0, otherwise

If X ~ U [0, 1], then

f(x) = 1, 0 ≤ x ≤ 1

f(x) = 0, otherwise

Distribution Function

If X ~ U [a, b], then

F(x) = P(X ≤ x) = ∫_a^x f(t) dt = (x - a) / (b - a), a ≤ x ≤ b

F(x) = 0, x < a

F(x) = 1, x > b

Mean and Variance

If X ~ U [a, b], then

Mean, E(X) = (a + b) / 2

Variance, Var(X) = (b - a)² / 12

Third Central Moment (μ₃)

Note μ₃ = 0, hence uniform distribution is symmetric. It is also evident from the nature of the graph of the probability density function.

Properties

1. U(a, b) is symmetric around its mean.
2. Since the probability density function graph is flat, there is no mode.
3. Median = Mean = (a + b) / 2

Results

Result 1: If X ~ U(a, b) then Y = (X - a) / (b - a) ~ U(0, 1).

Proof: Let X ~ U(a, b) with p.d.f. f(x) = 1 / (b - a), a ≤ x ≤ b. The p.d.f. of Y is also uniform over [0, 1].

Result 2: If X ~ U(a, b) then V = (b - X) / (b - a) ~ U(0, 1). The proof is similar to Result 1.

Result 3: If X is a continuous r.v. with p.d.f. f(x) and c.d.f. F(x) then Y = F(X) ~ U(0, 1).

Moment Generating Function (M.G.F.)

If X ~ U(a, b), the M.G.F. of X is given by:

M_X(t) = (e^bt - e^at) / (t(b - a))

Solved Examples

Example 3.1

If mean and variance of a U[a, b] r.v. are 5 and 3 respectively, determine the values of a and b.

Solution: If X ~ U[a, b] then E(X) = (a + b) / 2 and Var(X) = (b - a)² / 12

a + b = 10

(b - a)² / 12 = 3

b - a = 6

Solving these equations, we get a = 2, b = 8.

Example

On X'mas, John gives a party to his friends. A machine fills the ice cream cups. The quantity of ice cream per cup is uniformly distributed over 200 gms to 250 gms. (i) What is the probability that a friend of John gets a cup with more than 230 gms of ice cream? (ii) If in all twenty-five people attended the party and each had two cups of ice cream, what is the expected quantity of ice cream consumed in the party?

Solution: X = Quantity of ice cream per cup.

X ~ U [200, 250]

(i) P(X > 230) = 1 - P(X ≤ 230) = 1 - (230 - 200) / (250 - 200) = 0.4

(ii) E(X) = (200 + 250) / 2 = 225 gms

On average, the quantity per cup is 225 gms.

Total quantity consumed = 225 x 2 x 25 = 11250 gms = 11.25 kg.

Applications of Uniform Distribution

1. Although it is the simplest continuous distribution, it has wide applicability in research, mainly used as a prior model for parameters in Bayes' theory.
2. It is used to represent the distribution of rounding-off errors.
3. It is also used in life testing and traffic flow experiments.
4. If a r.v. Y follows any continuous distribution, then its distribution function X = F(Y) can be shown to follow U(0, 1). This important result facilitates sampling from any continuous distribution.

Note: The distribution has very striking properties, such as the distribution of the sum of two uniform r.v's is not uniform.

Discrete Distributions

• Degenerate
• Bernoulli
• Binomial Distribution
• Hypergeometric Distribution
• Poisson Distribution
• Geometric Distribution
• Negative Binomial Distribution
• Truncated Distribution

Continuous Distributions

• Continuous Univariate Distributions
• Continuous Bivariate Distributions
• Continuous Uniform Distribution
• Normal Distribution
• Exponential Distribution
• Gamma Distribution
• Log-normal Distribution
• Beta Distribution of First Kind
• Beta Distribution of Second Kind
• Laplace Distribution
• Weibull Distribution
• Cauchy Distribution

Sampling Distribution

• Chi-Square Distribution
• Student's t-distribution
• Snedecor's F-distribution
• Testing of Hypothesis