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3.2.1 Uniform Distribution

Overview

This section explores the uniform random variable, characterized by equal likelihood of assuming any value within a specific range. We'll look at its probability density function (PDF), cumulative distribution function (CDF), and how these concepts are applied to calculate probabilities and other statistical measures.

Key Concepts

  • Uniform Random Variable: A random variable that can take any value between a specified lower (L) and upper limit (U) with equal probability.
  • Probability Density Function (PDF): Constant across the interval [L, U] and zero elsewhere.
  • Cumulative Distribution Function (CDF): Provides the probability that the random variable is less than or equal to a certain value.

Detailed Explanation

1. Definition and Characteristics

  • Uniform Distribution: Defined by the probability being proportional to the length of the interval between L and U.
  • Mathematical Formulation:

  • PDF: image

  • CDF: image

  • Diagram Recommended: Show the flat shape of the PDF and the linear growth of the CDF within the interval.

Expectation and Variance

  • Expectation (Mean): image

  • Variance: image

  • Example and Calculation: Use a practical example, such as driving times, to calculate mean and variance.

Practical Example

  • Context: Nilesh Shah's driving time from Vadodara to Ahmedabad.
  • Application:
  • Expected driving time based on uniform distribution.
  • Probabilities of driving times within specific intervals.
  • Calculations:
  • Probability of driving time being less than 130 minutes.
  • Probability of driving time exceeding 105 minutes.
  • Special Note: Highlight that the probability of any precise value, like exactly 120 minutes, is zero in continuous distributions.