Discrete Random Variables: The discrete uniform distribution

Discrete Random Variables: The discrete uniform distribution

Discrete Random Variables

  • A discrete random variable can only take on a finite or countable number of values.
  • Each value has an associated probability, defining how likely that outcome is to occur.

Discrete Uniform Distribution

  • The discrete uniform distribution is a type of probability distribution in which all outcomes are equally likely.
  • A simple and common example of a discrete uniform distribution is a fair six-sided die. Each face (outcome) is equally likely to occur on each throw.

Characteristics of a Discrete Uniform Distribution

  • All outcomes are equally likely; the probability of any specific outcome is 1/n, where n is the number of outcomes.
  • A discrete uniform distribution is defined by the smallest and largest values in the range of possible outcomes.
  • The outcomes must be finite and countable, and each outcome should be independent of the others.

Parameters of the Discrete Uniform Distribution

  • A discrete uniform distribution is typically denoted as U(a, b) or DU(a, b), where a and b are the smallest and largest possible outcomes respectively.
  • The expected value (mean) of a discrete uniform distribution is (a + b)/2.
  • The variance is (b - a + 1)² - 1/12, and the standard deviation is the square root of the variance.

Probability Mass Function (PMF)

  • For a discrete uniform distribution, the probability mass function assigns an equal probability 1/n to each outcome.
  • With n outcomes, the PMF is P(X = k) = 1/n for a ≤ k ≤ b. ‘k’ is a specific outcome in the distribution’s range.

Cumulative Distribution Function (CDF)

  • For a discrete random variable, the cumulative distribution function gives the probability that the variable can take a value less than or equal to a certain value.
  • For a discrete uniform distribution, the CDF is F(k) = (k - a + 1) / (b - a + 1) for a ≤ k ≤ b.

Application of Discrete Uniform Distribution

  • Discrete Uniform Distribution is used in scenarios where we have a finite number of outcomes and each of these outcomes is equally likely to occur.