Glossary

5.2 The Uniform Distribution

3 min read•june 27, 2024

The assumes equal likelihood for all values within a set . It's used to model situations where any outcome in a given is equally probable, like waiting times or random selections.

Calculating probabilities for uniform distributions involves simple formulas based on the interval's endpoints. Key applications include quality control, , and modeling various real-world scenarios with .

The Uniform Distribution

Uniform distribution probability calculations

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  • Continuous probability distribution assumes equal likelihood for all values within a specified interval (also known as )
    • Denoted as X∼U(a,b)X \sim U(a, b), where aa represents the minimum value and bb represents the maximum value
  • () for the uniform distribution:
    • f(x)=1b−af(x) = \frac{1}{b-a} for values of xx falling within the interval [a,b][a, b]
    • f(x)=0f(x) = 0 for values of xx less than aa or greater than bb
  • for the uniform distribution:
    • F(x)=0F(x) = 0 for values of xx less than aa
    • F(x)=x−ab−aF(x) = \frac{x-a}{b-a} for values of xx within the interval [a,b][a, b]
    • F(x)=1F(x) = 1 for values of xx greater than bb
  • Calculate the probability of an event occurring within a specified interval [c,d][c, d] using the formula:
    • P(c≤X≤d)=d−cb−aP(c \leq X \leq d) = \frac{d-c}{b-a}, where cc and dd fall within the interval [a,b][a, b]
    • Example: The waiting time for a bus follows a uniform distribution between 5 and 15 minutes. The probability of waiting between 8 and 12 minutes is 12−815−5=0.4\frac{12-8}{15-5} = 0.4 or 40%

Equal likelihood in uniform distributions

  • Uniform distribution assumes all values within the specified interval [a,b][a, b] have the same probability of occurring ()
    • The probability of any specific value within the interval is zero due to the infinite number of possible values
  • The probability of an event occurring within a of [a,b][a, b] is proportional to the length of the subinterval
    • Longer subintervals have a higher probability of containing the event compared to shorter subintervals
    • Example: A dart thrown at a circular target has an equal likelihood of landing anywhere within the target area
  • Uniform distribution models situations where all outcomes within a given range are
    • Examples include the random selection of a card from a well-shuffled deck or the position of a randomly dropped object on a table

Key characteristics of uniform distributions

  • Range: The interval [a,b][a, b] represents the range of possible values for the uniform distribution
  • density: The PDF of a uniform distribution is constant throughout its range
  • The probability of an event occurring within any subinterval of equal length is the same

Applications of uniform distributions

  • Identify the minimum (aa) and maximum (bb) values of the distribution when solving problems using the uniform distribution
  • Determine the inclusivity or exclusivity of the interval endpoints
    • Inclusive endpoints denoted by square brackets [a,b][a, b] include aa and bb in the interval
    • Exclusive endpoints denoted by parentheses (a,b)(a, b) exclude aa and bb from the interval
  • Apply the appropriate probability density function (PDF) or (CDF) to calculate the desired probability
    • Use the correct formula based on the inclusivity or exclusivity of the endpoints
    • Example: The height of a randomly selected plant follows a uniform distribution between 20 and 30 cm. The probability of selecting a plant with a height less than 25 cm is F(25)=25−2030−20=0.5F(25) = \frac{25-20}{30-20} = 0.5 or 50%
  • Interpret the results in the context of the real-world problem
    • State the probability of the event occurring within the given interval and explain its significance
    • Example: In a quality control process, the weight of a product follows a uniform distribution between 95 and 105 grams. If a product weighs less than 98 grams, it is considered defective. The probability of a product being defective is 98−95105−95=0.3\frac{98-95}{105-95} = 0.3 or 30%

Key Terms to Review (24)

Constant Probability: Constant probability refers to a situation where the probability of an event occurring remains the same across multiple trials or observations. This concept is particularly relevant in the context of the binomial distribution and the uniform distribution.
Constant Probability Density: Constant probability density refers to a uniform distribution where the probability of a random variable occurring within a given interval is equal across the entire range of the distribution. This means the probability density function is constant, or flat, over the defined interval.
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 where all values within the given range are equally probable.
Cumulative Distribution Function: The cumulative distribution function (CDF) is a fundamental concept in probability theory and statistics that describes the probability of a random variable taking a value less than or equal to a given value. It is a function that provides the cumulative probability distribution of a random variable, allowing for the calculation of probabilities for various ranges of values.
Cumulative Distribution Function (CDF): The Cumulative Distribution Function (CDF) is a fundamental concept in probability and statistics that describes the probability of a random variable taking a value less than or equal to a specified value. It provides a comprehensive understanding of the distribution of a random variable and is widely used in various statistical analyses, including the Poisson distribution, uniform distribution, and exponential distribution.
Discrete Uniform Distribution: The discrete uniform distribution is a probability distribution where a discrete random variable can take on any value within a specified range of equally likely outcomes. It is characterized by a constant probability for each possible value within the range.
Equally Likely: Equally likely refers to a situation where all possible outcomes of an event have the same probability of occurring. This concept is fundamental to the understanding of the Uniform Distribution, which assumes that all outcomes within a given range have an equal chance of being observed.
Equiprobable Distribution: An equiprobable distribution is a probability distribution where all possible outcomes or events have an equal chance of occurring. In other words, each outcome has the same probability of being observed or selected.
F(x) = 1 / (b - a): The function f(x) = 1 / (b - a) is a key concept in the context of the Uniform Distribution. It represents the probability density function (PDF) of the Uniform Distribution, which describes the probability of a random variable X taking on a value within a specified range [a, b].
Interval: An interval is a range of values or a continuous segment on a numerical scale. It represents the distance between two points or the space occupied by a set of related values. Intervals are fundamental concepts in statistics, probability, and various other mathematical disciplines.
Lower Bound: The lower bound refers to the smallest possible value or limit that a variable or parameter can take within a given context. It represents the minimum or the lower limit of a range or distribution, and is an important concept in various statistical and mathematical analyses.
Mean: The mean, also known as the arithmetic mean or average, is a measure of central tendency that represents the central or typical value in a dataset. It is calculated by summing all the values in the dataset and dividing by the total number of values. The mean is a widely used statistic that provides information about the location or central tendency of a distribution.
Median: The median is a measure of the central tendency of a dataset, representing the middle value when the data is arranged in numerical order. It is a key statistical concept that provides information about the location and distribution of data points.
PDF: PDF, or Portable Document Format, is a file format developed by Adobe Systems to create and share documents that maintain their original formatting and layout across different computer systems and platforms. It is a widely-used standard for electronic document exchange, allowing for the secure and reliable distribution of information.
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 distribution of a continuous random variable.
Random Number Generation: Random number generation is the process of producing a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance. It is a fundamental concept in statistics and is crucial for simulations, cryptography, and various other applications that require unpredictable and unbiased data.
Range: The range is a measure of the spread or dispersion of a set of data. It is calculated as the difference between the largest and smallest values in the dataset, providing a simple way to quantify the variability or spread of the data.
Rectangular Distribution: The rectangular distribution, also known as the uniform distribution, is a continuous probability distribution where the random variable has an equal likelihood of taking on any value within a specified range. This distribution is characterized by a constant probability density function over the defined interval, making it a useful model for scenarios where all outcomes within a range are equally likely.
Subinterval: A subinterval is a smaller interval that is contained within a larger interval. It is a fundamental concept in the context of the uniform distribution, which describes a continuous random variable that is equally likely to take on any value within a specified interval.
Uniform Distribution: The uniform distribution is a continuous probability distribution where the probability of any outcome within a specified range is equally likely. It is characterized by a constant probability density function over a defined interval.
Uniform Probability: Uniform probability refers to a probability distribution where all outcomes in a sample space are equally likely to occur. This means that each possible outcome has the same probability of being observed, and the probability of any one outcome is the reciprocal of the total number of possible outcomes.
Uniform Random Variable: A uniform random variable is a continuous probability distribution where the random variable can take on any value within a specified range with equal likelihood. This type of distribution is characterized by a constant probability density function over the defined interval, indicating that all values within the range are equally probable.
Upper Bound: The upper bound is the maximum value or limit of a range or distribution. It represents the highest possible value that a variable or statistic can take on within a given context or scenario.
Variance: Variance is a statistical measure that quantifies the amount of variation or dispersion in a dataset. It represents the average squared deviation from the mean, providing a way to understand the spread or distribution of data points around the central tendency.
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