# Intensity of Exponential Distribution

How do I show the following: Suppose $\lambda=-\frac{S'(x)}{S(x)}$, where $S(x)=1-F(x)$ is survival probability. Show that $\lambda$ is the intensity of the exponential distribution with cdf $F(x)=1-e^{-\lambda x}$.

• I think the way the question is worded is awkward. I would have said: Suppose the intensity function defined by $\lambda(x)=\frac{S^{'}(x)}{S(x)}$ is equal to a constant $\lambda$. Show that in this case the cdf of the distribution is the cdf of an exponential, which is $F(x)=1-e^{-\lambda x}$. And you verify this by plugging in the $F(x)$ in the previous expression and seeing that you do indeed get a constant. – Alex C Jul 16 '17 at 17:40

## 1 Answer

You didn't define an intensity function, and you might be assuming what's to be shown. Does this help? \begin{align*} -\frac{S'(x)}{S(x)} &= \lambda\frac{e^{-\lambda x}}{e^{-\lambda x} } \tag{defn of survival function} \\ &= 1. \end{align*}