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exponential_distribution
##### Integrating the Exponential Distribution

$$\int e^{-x} \mbox{d}x = -e^{-x} + C$$ where $C$ is a constant.

Plugging in numbers from zero to very large we can see that $-e^{-x}$ ranges from $-1$ to $0$. By definition the area of the probability distribution has to be one (from zero to one when integrating from zero to infinity) so $C=1$.

This gives a Cumulative Density Function of $$\mbox{CDF}=1-e^{-x}$$

For example, 95% of the distribution lies below $x \approx 3$. $$0.95 = 1-e^{-x}$$ $$x \approx 2.996$$ and 99% of the distribution is above $x \approx 0.01$ $$0.01 = 1-e^{-x}$$ $$x \approx 0.01005$$