# Is there a geometric concept describing this?

$$\frac{\int_{t+h}^{\infty} \lambda e^{-\lambda x} dx}{\int_{t}^{\infty} \lambda e^{-\lambda x} dx} = \frac{\int_{h}^{\infty} \lambda e^{-\lambda x} dx}{\int_{0}^{\infty} \lambda e^{-\lambda x} dx}$$ 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?

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I have seen it called scale invariance. – Ross Millikan Dec 4 '12 at 14:44
Thanks. How is it exactly defined, and how do you judge whether a curve is scale-invariant? – qed Dec 4 '12 at 14:49
It 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. – Ross Millikan Dec 4 '12 at 15:53

$$\frac{\int_{t+h}^{\infty} e^{-\lambda x} dx}{\int_{t}^{\infty} e^{-\lambda x} dx} =\frac{[-\frac{1}{\lambda}e^{-\lambda x}]_{t+h}^{\infty}}{[-\frac{1}{\lambda}e^{-\lambda x}]_{t}^{\infty}} =\frac{e^{-\lambda (t+h)}}{e^{-\lambda t}} =e^{-\lambda h} = \frac{\int_{h}^{\infty} e^{-\lambda x} dx}{\int_{0}^{\infty} e^{-\lambda x} dx}$$