Find the probability of $prob(x\ge 4| x\ge1)$ using the exponential distribution.

Question

Find the probability of $prob(x\ge 4| x\ge1)$ using the exponential distribution.

$f(x)=\lambda e^{-\lambda x}$ for $x\ge0$ is the exponetial distribution

S0 $prob(x\ge 4| x\ge1)=prob(x\ge 3)$ by the memory less property. and then I would do

$P(x\ge3)=\int_{3}^{\infty}\lambda e^{-\lambda x)}$

and I get

$=\lambda e^{-3\lambda}$ but would this be right

Answer 1

$$ \mathbb{P}(X\ge 4| X\ge 1 )=\frac{\mathbb{P}(X\ge 4 \cap X \ge 1)}{\mathbb{P}(X\ge 1 )} = \frac{\mathbb{P}(X\ge 4)}{\mathbb{P}(X\ge 1 )} = \frac{e^{-\lambda4}}{e^{-\lambda}}= e^{-\lambda 3}. $$