what is the intuition for the variance of poisson distribution being lambda

Question

poisson distribution: $P(X=x)=\frac{\lambda^x e^{−x}}{x!}$,

It is easy to understand the mean the distribution equal to $\lambda$, but what is the intuition for the variance being equal to $\lambda$?

Answer 1

The variance of the sum of independent random variables is the sum of their variances. The Poisson distribution is the limit of a binomial distribution in which the number of trials goes to infinity while the expected total number of successes is held constant at $\lambda$. Divide a (say, unit) time interval into $n$ subintervals and consider each interval to contain a success with probability $\frac\lambda n$. The corresponding binomial distribution is the sum of $n$ Bernoulli distributions, each with expectation $\frac\lambda n$ and variance $\frac\lambda n\left(1-\frac\lambda n\right)$, so the expectation of the sum is $\lambda$ and the variance of the sum is $\lambda\left(1-\frac\lambda n\right)$. As $n\to\infty$, the variance goes to $\lambda$.