Questions: Uniform Distribution: Theory and Applications

Question 1

A computer can generate uniform random numbers U ~ Uniform[0,1] and needs to produce samples from an exponential distribution with CDF F(x) = 1 − e^(−λx). Which method works?

• A. Square U to approximate the exponential distribution's right-skewed shape
• B. Generate many uniform values and average them; by the CLT this converges to the exponential
• C. Compute X = F⁻¹(U) = −ln(1 − U)/λ; by the probability integral transform X follows the exponential distribution
• D. The exponential distribution cannot be derived from uniform random numbers without specialized hardware

Answer: (C)

This is inverse transform sampling. Since F(X) ~ Uniform[0,1] for any continuous RV X, running it in reverse: if U ~ Uniform[0,1], then X = F⁻¹(U) has CDF F. Applying the inverse exponential CDF to a uniform random number yields an exponential random variable. Options A and B are wrong approaches; option D is false — uniform numbers are the universal raw material for all continuous distributions.

Question 2

For a Uniform[a, b] distribution, doubling the interval width (b − a) changes the variance by a factor of:

• A. 2 — variance doubles when width doubles
• B. 4 — variance quadruples because Var = (b − a)²/12 and doubling the width squares to four times
• C. √2 — variance grows by the square root of the width increase
• D. The variance does not change — it depends only on the height of the density function, which is 1/(b − a)

Answer: (B)

Var(X) = (b − a)²/12. If (b − a) doubles to 2(b − a), the new variance is (2(b − a))²/12 = 4(b − a)²/12 — four times the original. The quadratic relationship between width and variance means width changes have an amplified effect on spread. Option D confuses the density height with the variance formula.

Question 3

For a Uniform[a, b] distribution, the probability that X lies in any sub-interval [c, d] ⊆ [a, b] is proportional to the length of that sub-interval.

• T. True
• F. False

Answer: True

Because f(x) = 1/(b − a) is constant, P(c ≤ X ≤ d) = (d − c)/(b − a), which is exactly the fraction of the total interval length occupied by [c, d]. This geometric interpretation is the key feature of the uniform distribution: probability equals relative length, and no sub-region is favored over any other of equal width.

Question 4

A large random sample from a Uniform[0, 1] distribution will have its values concentrated near the mean 0.5, just as a normal distribution concentrates near its mean.

• T. True
• F. False

Answer: False

The defining feature of a uniform distribution is that all sub-intervals of equal length have equal probability — there is no concentration anywhere. Values are spread evenly across the entire interval, not piled up at the center. This is precisely what distinguishes it from the normal distribution. Confusing these two is a common error that comes from overgeneralizing the 'bell curve' intuition.

Question 5

What is the probability integral transform, and why does it make the uniform distribution the universal foundation of random simulation?

Model answer: The probability integral transform states: if X is any continuous random variable with CDF F, then U = F(X) ~ Uniform[0, 1]. Running this in reverse — if U ~ Uniform[0, 1], then X = F⁻¹(U) has CDF F — is the foundation of inverse transform sampling. To generate a random draw from any continuous distribution, you only need to: (1) generate a uniform pseudo-random number U, and (2) apply the inverse CDF. Every software function that draws from a normal, exponential, beta, or gamma distribution ultimately starts from uniform pseudo-random numbers and transforms them this way. The uniform distribution is thus the raw material from which all other continuous distributions are constructed in simulation.

This is the deepest result in the topic. Students who know only the mean and variance formula have memorized facts; students who understand the probability integral transform understand why the uniform distribution occupies a foundational role in all of probability and statistics.