How do you solve for lambda in an exponential distribution?

Question

I am working on a problem and am unsure how to solve it.

The problem: Find an exponential distribution such that P(Z $\ge$ 3) = .04

What I have done so far:

P(Z$\ge$3) = 1 - P(Z$\lt$ 3)

We are solving for $\lambda$ in X ~ Exp($\lambda$)

Quantile is 4% (I think)

What steps do I need to do to solve this problem?

Answer 1

We have

\begin{align} \mathbb{P}(Z \geq 3) = \int_3^{\infty} \lambda e^{-\lambda x} dx = [-e^{-\lambda x}]_3^{\infty} = e^{-3\lambda}, \end{align}

hence $\mathbb{P}(Z \geq 3) = 0.04$ iff $\lambda = \frac{\log(0.04)}{-3}$.