Deriving Exponential Distribution Intensity from Survival Probability

Summary

The question asks how the intensity, or hazard rate, follows from a survival function defined as one minus the cumulative distribution. For an exponential distribution with rate parameter λ, the cumulative distribution is one minus the exponential survival probability. Differentiating that survival function and dividing its negative by the survival probability yields a constant rate, λ.

The included answer challenges the prompt for not defining intensity, but its displayed calculation is incomplete and reaches an incorrect intermediate result: it omits the derivative of the exponential survival function. The intended identity is straightforward once that derivative is included. The note provides no broader discussion of hazard rates, assumptions, or applications, so it serves mainly as a concise calculus relationship and a reminder to state definitions precisely.

Key ideas

• The survival function is the probability that a random variable exceeds a given value.
• For an exponential distribution, survival probability decays exponentially at a constant rate.
• The negative derivative of survival probability divided by survival probability equals the hazard rate.
• The displayed answer omits the derivative and does not correctly demonstrate the stated identity.

Tags

# Intensity of Exponential Distribution


# 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 $F(x)=1-e^{-\lambda x}$.

## Answer by Taylor (score 0)

https://quant.stackexchange.com/a/35142

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*}

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.