$ \frac{\int_{t+h}^{\infty} \lambda e^{-\lambda x} dx}{\int_{t}^{\infty} \lambda e^{-\lambda x} dx} = \frac{\int_{h}^{\infty} \lambda e^{-\lambda x} dx}{\int_{0}^{\infty} \lambda e^{-\lambda x} dx} $ This is known as the memoryless property of the exponential distribution. From the plots we can see the shape of the curve looks the same wherever you start plotting. Is there a concept describing this?
Is there a geometric concept describing this?
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0It is defined precisely by being an exponential or geometric distribution. The motivation is just what you have seen-the curve looks the same at all scales. – 2012-12-04
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$ \frac{\int_{t+h}^{\infty} e^{-\lambda x} dx}{\int_{t}^{\infty} e^{-\lambda x} dx} =\frac{[-\frac{1}{\lambda}e^{-\lambda x}]_{t+h}^{\infty}}{[-\frac{1}{\lambda}e^{-\lambda x}]_{t}^{\infty}} =\frac{e^{-\lambda (t+h)}}{e^{-\lambda t}} =e^{-\lambda h} = \frac{\int_{h}^{\infty} e^{-\lambda x} dx}{\int_{0}^{\infty} e^{-\lambda x} dx} $
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0@MartinSleziak thanks for pointing that out. I added the link to the tag. Sorry for the inconvience. – 2012-12-10