Pareto PDF differentiation gives a negative CDF

Question

I am trying to calculate the CDF based on the PDF for the Pareto distribution, I've done the differentiation but I end up with a negative functionlike: $-(\frac{\lambda}{(x+\lambda)})^\alpha$

The PDF is: $\alpha\lambda^\alpha(\lambda+x)^{-\alpha-1}$

Even WolframAlpha gives me that

How is it possible that the CDF is negative? isn't it supposed to be the area under the curve?

Answer 1

With your parameters and the integral $$I(x) = \int \alpha \lambda^\alpha(\lambda+x)^{-\alpha-1} dx = -\frac{\lambda^\alpha}{(x+\lambda)^\alpha}$$ you get the Pareto CDF $F(x)$ as $$F(x) = \int_0^x \alpha \lambda^\alpha(\lambda+t)^{-\alpha-1} dt = I(x)-I(0)= 1 - \frac{\lambda^\alpha}{(x+\lambda)^\alpha}$$ which is valid positive CDF for $\alpha >0, \lambda > 0.$ Wolfram Alpha computes integrate a*l^a(l+x)^(-a-1) x=0..infinity $$\int_0^\infty \alpha \lambda^\alpha(\lambda+t)^{-\alpha-1} dt = 1$$

Answer 2

I think you mean you integrated. You forgot the integration constant, which by a definite integral is $1$.