Question 1

What is the uniform distribution?

Accepted Answer

The continuous uniform distribution U(a, b) models a random variable with equal probability density everywhere in [a, b] and zero probability outside that range. Examples: rounding error in numerical computation (U(−0.5, 0.5)), the position of a randomly thrown dart on a line, or a random number between 0 and 1 from a computer. The "rectangular" shape of its PDF gives it the alternative name rectangular distribution.

Question 2

What is the mean and variance of the uniform distribution?

Accepted Answer

Mean = (a + b) / 2 — exactly the midpoint of the interval. Variance = (b − a)² / 12 — wider intervals have more variance. Standard deviation = (b − a) / √12 ≈ (b − a) × 0.289. The median equals the mean (the midpoint). Skewness = 0 (perfectly symmetric). Excess kurtosis = −6/5 = −1.2 (lighter tails than normal — it is platykurtic).

Question 3

How do I compute P(x₁ ≤ X ≤ x₂) for a uniform distribution?

Accepted Answer

P(x₁ ≤ X ≤ x₂) = F(x₂) − F(x₁) = (x₂ − x₁) / (b − a). The probability is simply proportional to the length of the sub-interval relative to the total interval. For example, if X ~ U(0, 10), then P(3 ≤ X ≤ 7) = (7 − 3) / (10 − 0) = 4/10 = 0.4.

Question 4

What is the relationship between uniform and other distributions?

Accepted Answer

The uniform distribution is foundational in probability and simulation: (1) If U ~ U(0,1), then −ln(U)/λ ~ Exponential(λ). (2) If U ~ U(0,1), then F⁻¹(U) follows any distribution with CDF F (the inverse CDF method). (3) The sum of 12 U(0,1) values minus 6 approximates N(0,1). (4) U(0,1) is the basis of Monte Carlo simulation. Any random number generator effectively produces U(0,1) samples.

Question 5

When should I use the uniform distribution?

Accepted Answer

Use the uniform distribution when: (1) all outcomes in a range are equally likely (e.g., equally likely arrival time within a window), (2) you have maximum uncertainty and only know the bounds (maximum entropy principle), (3) modeling rounding errors in floating-point arithmetic, (4) as a non-informative prior in Bayesian statistics, or (5) generating random samples via the inverse CDF method for simulation.

Question 6

What is the discrete uniform distribution?

Accepted Answer

The discrete uniform distribution assigns equal probability 1/n to each of n distinct outcomes. For example, rolling a fair die: P(X = k) = 1/6 for k = 1, 2, 3, 4, 5, 6. Mean = (1 + n) / 2, Variance = (n² − 1) / 12. The continuous uniform distribution is the limit as the number of equally spaced discrete outcomes grows to infinity.

Property	Formula	Example U(0,10)
Support	[a, b]	[0, 10]
PDF f(x)	1 / (b−a)	0.1
Mean	(a+b) / 2	5
Median	(a+b) / 2	5
Variance	(b−a)² / 12	8.333
Std deviation	(b−a) / √12	2.887
Skewness	0 (symmetric)	0
Excess kurtosis	−6/5 = −1.2	−1.2

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