Glossary

5.2 The Uniform Distribution

3 min read•june 25, 2024

The is a where all values within a range have equal likelihood. It's defined by its minimum and maximum values, with a constant between them.

Understanding uniform distributions is crucial for modeling scenarios with equal probabilities. We'll explore how to calculate probabilities, work with inclusive and exclusive endpoints, and determine measures of central tendency and variability for uniform distributions.

The Uniform Distribution

Uniform distribution probability calculations

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  • (also known as ) is a continuous probability distribution where the probability of a value occurring remains constant for all values within a specified range
    • Denoted as X∼U(a,b)X \sim U(a, b), where aa represents the and bb represents the ()
  • Probability density function (PDF) for a uniform distribution is defined as:
    • f(x)=1b−af(x) = \frac{1}{b-a} for a≤x≤ba \leq x \leq b (constant probability within the range)
    • f(x)=0f(x) = 0 for x<ax < a or x>bx > b (zero probability outside the range)
  • Calculate the probability of an event within the range [a,b][a, b] using the formula:
    • P(c≤X≤d)=d−cb−aP(c \leq X \leq d) = \frac{d-c}{b-a}, where a≤c<d≤ba \leq c < d \leq b (probability is proportional to the length of the )
  • (CDF) for a uniform distribution is defined as:
    • F(x)=0F(x) = 0 for x<ax < a (zero probability below the minimum value)
    • F(x)=x−ab−aF(x) = \frac{x-a}{b-a} for a≤x≤ba \leq x \leq b (probability increases linearly within the range)
    • F(x)=1F(x) = 1 for x>bx > b (certain event above the maximum value)

Equal likelihood in uniform distributions

  • In a uniform distribution, all values within the specified range have an equal probability of occurring
    • PDF remains constant over the entire range (flat probability density)
  • Probability of a specific value occurring within the range is 0, as there are infinitely many values in a continuous distribution
    • Focus on the probability of events occurring within subintervals of the range
  • Probability of an event occurring within a subinterval of the range is proportional to the length of the subinterval
    • Longer subintervals have a higher probability of containing the event (proportional to subinterval length)

Uniform distributions with inclusive/exclusive endpoints

  • When solving problems involving uniform distributions, consider whether the endpoints of the given interval are inclusive or exclusive
    • Inclusive endpoints are denoted using square brackets [][ ], indicating the endpoint values are included in the interval ()
    • Exclusive endpoints are denoted using parentheses ()( ), indicating the endpoint values are not included in the interval ()
  • For inclusive endpoints, use the standard probability formula:
    • P(c≤X≤d)=d−cb−aP(c \leq X \leq d) = \frac{d-c}{b-a}, where a≤c≤d≤ba \leq c \leq d \leq b (probability includes both endpoints)
  • For exclusive endpoints, adjust the formula to exclude the endpoint values:
    • P(c<X<d)=d−cb−aP(c < X < d) = \frac{d-c}{b-a}, where a≤c<d≤ba \leq c < d \leq b (probability excludes both endpoints)
  • When using the CDF to solve problems, consider the endpoint conditions:
    • For inclusive lower endpoint and exclusive upper endpoint: P(c≤X<d)=F(d)−F(c)P(c \leq X < d) = F(d) - F(c) (probability between cc and dd, excluding dd)
    • For exclusive lower endpoint and inclusive upper endpoint: P(c<X≤d)=F(d)−F(c+)P(c < X \leq d) = F(d) - F(c^+), where c+c^+ represents a value just above cc (probability between cc and dd, excluding cc)

Measures of central tendency and variability

  • (mean) of a uniform distribution is the midpoint of the interval:
    • E(X)=a+b2E(X) = \frac{a + b}{2}
  • of a uniform distribution measures the spread of values:
    • Var(X)=(b−a)212Var(X) = \frac{(b-a)^2}{12}
  • is the square root of the variance:
    • SD(X)=Var(X)=b−a12SD(X) = \sqrt{Var(X)} = \frac{b-a}{\sqrt{12}}

Key Terms to Review (18)

Closed Interval: A closed interval is a set of real numbers that includes both the lower and upper endpoints. It is denoted using square brackets, such as [a, b], where 'a' represents the lower endpoint and 'b' represents the upper endpoint. The closed interval contains all the numbers between 'a' and 'b', including 'a' and 'b' themselves.
Continuous Probability Distribution: A continuous probability distribution is a type of probability distribution where the random variable can take on any value within a given range or interval, rather than being limited to discrete values. This type of distribution is used to model continuous phenomena, such as measurements or quantities that can vary smoothly and take on an infinite number of possible values.
Continuous Uniform Distribution: The continuous uniform distribution is a probability distribution that describes a random variable with an equal likelihood of taking on any value within a specified interval. It is a continuous probability function that models situations where all outcomes within a given range are equally likely to occur.
Cumulative Distribution Function: The cumulative distribution function (CDF) is a function that describes the probability that a random variable takes on a value less than or equal to a specific value. It provides a complete picture of the distribution of probabilities for both discrete and continuous random variables, enabling comparisons and insights across different types of distributions.
Equally likely: Equally likely events are those that have the same probability of occurring. In probability theory, this concept is fundamental when considering outcomes in a uniform distribution.
Expected Value: Expected value is a fundamental concept in probability that represents the long-term average or mean of a random variable's outcomes, weighted by their probabilities. It provides a way to quantify the center of a probability distribution and is crucial in decision-making processes involving risk and uncertainty.
Interval Endpoints: Interval endpoints refer to the specific values that define the boundaries of a statistical interval. These endpoints establish the range within which a variable or observation is expected to fall, and they are crucial in understanding and analyzing various probability distributions, such as the uniform distribution.
Maximum Value: The maximum value refers to the largest or highest possible value within a given set of data or distribution. It represents the upper bound or the greatest magnitude that a variable or observation can attain in a specific context.
Minimum Value: The minimum value is the smallest or lowest numerical value within a set of data or a distribution. It represents the lower bound or the smallest possible value that a variable or observation can take on in a given context.
Open Interval: An open interval is a set of real numbers that includes all numbers between two endpoints but does not include the endpoints themselves. This concept is crucial when discussing the uniform distribution, as it helps define the range of possible values for a random variable and indicates that values on the boundary are not considered part of the distribution.
Probability Density Function: The probability density function (PDF) is a mathematical function that describes the relative likelihood of a continuous random variable taking on a particular value. It provides a way to quantify the probability of a variable falling within a specified range of values.
Rectangular Distribution: The rectangular distribution, also known as the uniform distribution, is a probability distribution where the random variable has an equal chance of taking on any value within a specified range. This distribution is characterized by a constant probability density function over a finite interval, indicating that all values within the range are equally likely to occur.
Standard Deviation: Standard deviation is a statistic that measures the dispersion or spread of a set of values around the mean. It helps quantify how much individual data points differ from the average, indicating the extent to which values deviate from the central tendency in a dataset.
Subinterval: A subinterval is a smaller segment or portion of a larger interval. It represents a specific range within a given interval, allowing for more detailed analysis or examination of a particular section of the overall range.
U(a,b): U(a,b) represents a continuous random variable that follows a Uniform Distribution, where the variable can take on any value between the lower bound 'a' and the upper bound 'b'. The Uniform Distribution is a probability distribution that assigns an equal probability to all values within the specified range, making it a useful model for situations where all outcomes within a given interval are equally likely.
Uniform distribution: A uniform distribution is a type of probability distribution in which all outcomes are equally likely. In a continuous uniform distribution, every interval of the same length within the distribution's range has an equal probability of occurring.
Uniform Distribution: The uniform distribution is a continuous probability distribution where the probability of a random variable falling within a given interval is proportional to the length of the interval. This distribution is characterized by a constant probability density function over a specified range of values.
Variance: Variance is a statistical measurement that describes the spread or dispersion of a set of data points in relation to their mean. It quantifies how far each data point in the set is from the mean and thus from every other data point. A higher variance indicates that the data points are more spread out from the mean, while a lower variance shows that they are closer to the mean.
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