The uniform distribution assumes equal likelihood for all values within a set range. It's used to model situations where any outcome in a given interval 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, random number generation, and modeling various real-world scenarios with constant probability density.

The Uniform Distribution

Uniform distribution probability calculations

• Continuous probability distribution assumes equal likelihood for all values within a specified interval (also known as rectangular distribution)
  • Denoted as $X \sim U(a, b)$, where $a$ represents the minimum value and $b$ represents the maximum value
• Probability density function (PDF) for the uniform distribution:
  • $f(x) = \frac{1}{b-a}$ for values of $x$ falling within the interval $[a, b]$
  • $f(x) = 0$ for values of $x$ less than $a$ or greater than $b$
• Cumulative distribution function (CDF) for the uniform distribution:
  • $F(x) = 0$ for values of $x$ less than $a$
  • $F(x) = \frac{x-a}{b-a}$ for values of $x$ within the interval $[a, b]$
  • $F(x) = 1$ for values of $x$ greater than $b$
• Calculate the probability of an event occurring within a specified interval $[c, d]$ using the formula:
  • $P(c \leq X \leq d) = \frac{d-c}{b-a}$, where $c$ and $d$ fall within the interval $[a, b]$
  • 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 $\frac{12-8}{15-5} = 0.4$ or 40%

Equal likelihood in uniform distributions

• Uniform distribution assumes all values within the specified interval $[a, b]$ have the same probability of occurring (equiprobable distribution)
  • 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 subinterval of $[a, b]$ 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 equally likely
  • 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 $[a, b]$ represents the range of possible values for the uniform distribution
• Constant probability 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 ($a$) and maximum ($b$) 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 $[a, b]$ include $a$ and $b$ in the interval
  • Exclusive endpoints denoted by parentheses $(a, b)$ exclude $a$ and $b$ from the interval
• Apply the appropriate probability density function (PDF) or cumulative distribution function (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 $F(25) = \frac{25-20}{30-20} = 0.5$ 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 $\frac{98-95}{105-95} = 0.3$ or 30%