Exam Questions - Continuous uniform / rectangular distribution

Exam Questions - Continuous uniform / rectangular distribution

Continuous Uniform Distribution

  • A continuous uniform distribution, also known as a rectangular distribution, is a type of probability distribution where all intervals of the same length on the distribution’s support are equally probable.
  • Continuous uniform distribution is illustrated by a rectangle with constant height. This height is often denoted as ‘k’. The area of the rectangle, i.e., the total probability, should be equal to 1.

Deciphering questions on Uniform Distribution

  • Questions often provide you with the limits or bounds of the distribution, denoted as a and b. These represent the minimum and maximum possible values of the variable.
  • The height or ‘k’ can be calculated using the formula k = 1/(b - a).
  • Sometimes you are asked to find the mean or variance of a uniform distribution. The formula for the mean is μ = (a + b)/ 2, and for the variance is σ² = (b - a)² / 12.

Problem-solving Tips

  • Start by reading the question carefully and determining the minimum and maximum values.
  • Next, calculate the constant height ‘k’, followed by the mean and variance if required.
  • The range of the random variable forms an integral part of the solutions. In case of finding probabilities for a particular value, always sketch a graph to provide a visual aid.

Working with Cumulative Distribution Function

  • A Cumulative Distribution Function (CDF) for a random variable is defined as the probability that the variable takes a value less than or equal to a certain value.
  • For a uniform distribution, the CDF function if x < a is 0, if a ≤ x ≤ b then CDF is given by F(x) = (x - a) / (b - a), and if x > b then CDF is 1.
  • Exercises might involve comparing two distributions or creating a composite distribution.

Practical Applications

  • Understanding of uniform distributions can be useful in a variety of real-world contexts, including computer simulations, manufacturing, and quality control.
  • Recognising uniform distributions in these and other contexts can help to inherently understand the fairness and randomness implied in “uniform”, which is a key component in decision-making processes.