Exponential distribution from Poisson

In Poisson distribution, the probability of inter arrival time to be t or less is:

$$ P(X\leq t)= 1 - P(X>t) = 1 - P(0 \mbox{ arrivals in } t) = 1 - e^{-\lambda t} $$

and probability of one arrival in t is:

$$ P(k=1)= \lambda t e^{- \lambda t} $$

I wonder how the exponential distribution can be derived from Poisson to reach:

$$ P(X\leq t)= \lambda e^{- \lambda t} $$

Thanks,

Solution 1:

Suppose that $N = \{N_t: t \geq 0 \}$ is a Poisson process with rate $\lambda$. Let $X$ be the waiting time until the first occurrence in the process $N$. Then, for any $t > 0$, $$ {\rm P}(X \le t) = 1 - {\rm P}(X > t) = 1 - {\rm P}(N_t = 0) = 1 - e^{ - \lambda t}. $$ Hence, $X$ is an exponential random variable with density function $f_X$ given by $$ f_X {(t)} = \frac{{\rm d}}{{{\rm d}t}}(1 - e^{ - \lambda t} ) = \lambda e^{-\lambda t},\;\; t > 0. $$