Statistics - Uniforn Distribution

 
   
     
       

Uniform Distribution

       

Understanding the Uniform Distribution

       

          The Uniform Distribution is a probability distribution where all outcomes in an interval \([a, b]\) are equally likely. It is also called the rectangular distribution because of its constant probability.        

        Uniform Distribution Graph      
     
       
         

Key Metrics:

         
               
  • Mean: \( \mu = \frac{1}{2}(a + b) \)
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  • Variance: \( \sigma^2 = \frac{1}{12}(b - a)^2 \)
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Probability Density Function (PDF):

         

            \[             f(x) =             \begin{cases}               \frac{1}{b - a}, & a \leq x \leq b \\               0, & x < a \text{ or } x > b             \end{cases}             \]          

         

Cumulative Distribution Function (CDF):

         

            \[             F(x) =             \begin{cases}               0, & x < a \\               \frac{x - a}{b - a}, & a \leq x \leq b \\               1, & x \geq b             \end{cases}             \]          

         

Key Properties:

         
               
  • All values between \(a\) and \(b\) are equally likely.
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  • Graph is flat and horizontal over \([a, b]\).
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  • Minimum and maximum define the bounds of probability.
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  • It is a special case of the Beta distribution when \(\alpha = \beta = 1\).
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Applications:

         
               
  • Modeling scenarios with equal probabilities (e.g., rolling a fair die).
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  • Simulation and random number generation.
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  • Describing waiting time when all durations are equally probable.
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  • Used in computer algorithms for sampling or initialization.
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