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The shifted Gompertz distribution is the distribution of the largest order statistic of two independent random variables which are distributed exponential and Gompertz with parameters b and b and $$\eta$$ respectively. It has been used as a model of adoption of innovation.

Probability density function
The probability density function of the shifted Gompertz distribution is:


 * $$ f(x;b,\eta) = b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right] \mathrm{for}\ x > 0 \,\!$$

where $$b > 0$$ is the scale parameter and $$\eta > 0$$ is the shape parameter of the shifted Gompertz distribution.

Cumulative distribution function
The cumulative distribution function of the shifted Gompertz distribution is:


 * $$ F(x;b,\eta) = \left(1 - e^{-bx}\right)e^{-\eta e^{-bx}} \mathrm{for}\ x > 0 \,\!$$

Properties
The shifted Gompertz distribution is right-skewed for all values of $$\eta$$.

Shapes
The shifted Gompertz density function can take on different shapes depending on the values of the shape parameter $$\eta$$:
 * $$\eta \leq 0.5\,$$ the probability density function has mode 0.
 * $$\eta > 0.5\,$$ the probability density function has the mode at $$(-1/b)\ln(x^\star)\,, 0 < x^\star < 1$$ where $$x^\star\,$$ is the smallest root of $$\eta^2x^2 - \eta(3 + \eta)x + \eta + 1 = 0\,$$ which is $$x^\star = [3 + \eta - (\eta^2 + 2\eta + 5)^{1/2}]/(2\eta)$$

Related Distributions
If $$\eta$$ varies according to a gamma distribution with shape parameter $$\alpha$$ and scale parameter $$\beta$$ (mean = $$\alpha\beta$$), the cumulative distribution function is Gamma/Shifted Gompertz.