Comparing the Exponential Truncated Distribution with the Exponential Distribution











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I have the truncated exponential distribution: $$f(x)=frac{lambda e^{-lambda x}}{1-e^{-lambda k }} quad text{ for } 0 <x<k$$ The expected value of this for $k=frac{lambda}{5}$ is $$frac{1}{lambda}(1-frac{5}{e^5-1})$$ I'm trying to compare this expected value with the expected value of the exponential distribution with parameter $lambda$ and all I can see is that it's just smaller by a factor of $(1-frac{5}{e^5-1})$. I'm wondering if there's anything more that this relation between their expected values reveals other than the truncated exponential distribution has a smaller expected value.






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    I have the truncated exponential distribution: $$f(x)=frac{lambda e^{-lambda x}}{1-e^{-lambda k }} quad text{ for } 0 <x<k$$ The expected value of this for $k=frac{lambda}{5}$ is $$frac{1}{lambda}(1-frac{5}{e^5-1})$$ I'm trying to compare this expected value with the expected value of the exponential distribution with parameter $lambda$ and all I can see is that it's just smaller by a factor of $(1-frac{5}{e^5-1})$. I'm wondering if there's anything more that this relation between their expected values reveals other than the truncated exponential distribution has a smaller expected value.






    exponential-distribution







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      I have the truncated exponential distribution: $$f(x)=frac{lambda e^{-lambda x}}{1-e^{-lambda k }} quad text{ for } 0 <x<k$$ The expected value of this for $k=frac{lambda}{5}$ is $$frac{1}{lambda}(1-frac{5}{e^5-1})$$ I'm trying to compare this expected value with the expected value of the exponential distribution with parameter $lambda$ and all I can see is that it's just smaller by a factor of $(1-frac{5}{e^5-1})$. I'm wondering if there's anything more that this relation between their expected values reveals other than the truncated exponential distribution has a smaller expected value.






      exponential-distribution







      share|cite|improve this question













      I have the truncated exponential distribution: $$f(x)=frac{lambda e^{-lambda x}}{1-e^{-lambda k }} quad text{ for } 0 <x<k$$ The expected value of this for $k=frac{lambda}{5}$ is $$frac{1}{lambda}(1-frac{5}{e^5-1})$$ I'm trying to compare this expected value with the expected value of the exponential distribution with parameter $lambda$ and all I can see is that it's just smaller by a factor of $(1-frac{5}{e^5-1})$. I'm wondering if there's anything more that this relation between their expected values reveals other than the truncated exponential distribution has a smaller expected value.






      exponential-distribution




      exponential-distribution






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