Statistics - Exponential Distribution

 
   
     
       

Exponential Distribution

       

Understanding the Exponential Distribution in Statistics

       

          The Exponential Distribution is a continuous probability distribution used to model the time between independent events that occur at a constant average rate. It is widely applied in survival analysis, reliability engineering, and queuing systems.        

        Exponential Distribution Graph      
     
       
         

Key Formulas:

         
           

              \[               \mu = \frac{1}{\lambda}, \quad \sigma^2 = \frac{1}{\lambda^2}               \]              
              \[               f(x) = \lambda e^{-\lambda x}, \quad (\lambda > 0,\ x \geq 0)               \]              
              \[               F(x) = 1 - e^{-\lambda x}               \]            

         
         

Where:

         
               
  • \( \mu \): Mean (expected value) of the distribution
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  • \( \sigma^2 \): Variance of the distribution
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  • \( \lambda \): Rate parameter (average number of events per unit time)
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  • \( f(x) \): Probability density function (PDF)
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  • \( F(x) \): Cumulative distribution function (CDF)
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Key Properties:

         
               
  • Memoryless property: \( P(X > s + t \mid X > s) = P(X > t) \)
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  • Mean is the reciprocal of the rate: \( \mu = 1/\lambda \)
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  • Always non-negative: \( x \geq 0 \)
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  • Used when events occur continuously and independently
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Applications:

         
               
  • Modeling time until system/component failure
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  • Call center or server wait times
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  • Radioactive decay and other decay processes
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  • Survival analysis in medical statistics
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